Documentation

Mathlib.SetTheory.Cardinal.Basic

Cardinal Numbers #

We define cardinal numbers as a quotient of types under the equivalence relation of equinumerity.

Main definitions #

Main instances #

Main Statements #

Implementation notes #

References #

Tags #

cardinal number, cardinal arithmetic, cardinal exponentiation, aleph, Cantor's theorem, König's theorem, Konig's theorem

The equivalence relation on types given by equivalence (bijective correspondence) of types. Quotienting by this equivalence relation gives the cardinal numbers.

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def Cardinal :
Type (u + 1)

Cardinal.{u} is the type of cardinal numbers in Type u, defined as the quotient of Type u by existence of an equivalence (a bijection with explicit inverse).

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    The cardinal number of a type

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      The cardinal number of a type

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        theorem Cardinal.inductionOn {p : Cardinal.{u_1}Prop} (c : Cardinal.{u_1}) (h : ∀ (α : Type u_1), p (Cardinal.mk α)) :
        p c
        theorem Cardinal.inductionOn₂ {p : Cardinal.{u_1}Cardinal.{u_2}Prop} (c₁ : Cardinal.{u_1}) (c₂ : Cardinal.{u_2}) (h : ∀ (α : Type u_1) (β : Type u_2), p (Cardinal.mk α) (Cardinal.mk β)) :
        p c₁ c₂
        theorem Cardinal.inductionOn₃ {p : Cardinal.{u_1}Cardinal.{u_2}Cardinal.{u_3}Prop} (c₁ : Cardinal.{u_1}) (c₂ : Cardinal.{u_2}) (c₃ : Cardinal.{u_3}) (h : ∀ (α : Type u_1) (β : Type u_2) (γ : Type u_3), p (Cardinal.mk α) (Cardinal.mk β) (Cardinal.mk γ)) :
        p c₁ c₂ c₃
        theorem Cardinal.eq {α : Type u} {β : Type u} :
        @[simp]
        theorem Cardinal.mk'_def (α : Type u) :
        α = Cardinal.mk α

        The representative of the cardinal of a type is equivalent to the original type.

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          theorem Cardinal.mk_congr {α : Type u} {β : Type u} (e : α β) :
          theorem Equiv.cardinal_eq {α : Type u} {β : Type u} (e : α β) :

          Alias of Cardinal.mk_congr.

          def Cardinal.map (f : Type u → Type v) (hf : (α β : Type u) → α βf α f β) :

          Lift a function between Type*s to a function between Cardinals.

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            @[simp]
            theorem Cardinal.map_mk (f : Type u → Type v) (hf : (α β : Type u) → α βf α f β) (α : Type u) :
            def Cardinal.map₂ (f : Type u → Type v → Type w) (hf : (α β : Type u) → (γ δ : Type v) → α βγ δf α γ f β δ) :

            Lift a binary operation Type* → Type* → Type* to a binary operation on Cardinals.

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              The universe lift operation on cardinals. You can specify the universes explicitly with lift.{u v} : Cardinal.{v} → Cardinal.{max v u}

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                @[simp]

                lift.{max u v, u} equals lift.{v, u}.

                @[simp]

                lift.{max v u, u} equals lift.{v, u}.

                @[simp]

                A cardinal lifted to a lower or equal universe equals itself.

                @[simp]

                A cardinal lifted to the same universe equals itself.

                A cardinal lifted to the zero universe equals itself.

                We define the order on cardinal numbers by #α ≤ #β if and only if there exists an embedding (injective function) from α to β.

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                • One or more equations did not get rendered due to their size.
                theorem Cardinal.le_def (α : Type u) (β : Type u) :
                theorem Cardinal.mk_le_of_injective {α : Type u} {β : Type u} {f : αβ} (hf : Function.Injective f) :
                theorem Function.Embedding.cardinal_le {α : Type u} {β : Type u} (f : α β) :
                theorem Cardinal.mk_le_of_surjective {α : Type u} {β : Type u} {f : αβ} (hf : Function.Surjective f) :
                theorem Cardinal.le_mk_iff_exists_set {c : Cardinal.{u}} {α : Type u} :
                c Cardinal.mk α ∃ (p : Set α), Cardinal.mk p = c
                theorem Cardinal.mk_set_le {α : Type u} (s : Set α) :
                @[simp]
                theorem Cardinal.mk_preimage_down {α : Type u} {s : Set α} :

                A variant of Cardinal.lift_mk_le with specialized universes. Because Lean often can not realize it should use this specialization itself, we provide this statement separately so you don't have to solve the specialization problem either.

                A variant of Cardinal.lift_mk_eq with specialized universes. Because Lean often can not realize it should use this specialization itself, we provide this statement separately so you don't have to solve the specialization problem either.

                @[simp]
                theorem Cardinal.mk_eq_zero (α : Type u) [IsEmpty α] :
                @[simp]
                theorem Cardinal.mk_ne_zero (α : Type u) [Nonempty α] :
                theorem Cardinal.mk_eq_one (α : Type u) [Unique α] :

                Alias of the reverse direction of Cardinal.mk_le_one_iff_set_subsingleton.

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                theorem Cardinal.add_def (α : Type u) (β : Type u) :
                @[simp]
                @[simp]
                theorem Cardinal.mk_fintype (α : Type u) [h : Fintype α] :
                theorem Cardinal.cast_succ (n : ) :
                (n + 1) = n + 1
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                theorem Cardinal.mul_def (α : Type u) (β : Type u) :

                The cardinal exponential. #α ^ #β is the cardinal of β → α.

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                theorem Cardinal.power_def (α : Type u) (β : Type u) :
                @[simp]
                theorem Cardinal.power_zero {a : Cardinal.{u_1}} :
                a ^ 0 = 1
                @[simp]
                theorem Cardinal.power_one {a : Cardinal.{u}} :
                a ^ 1 = a
                theorem Cardinal.power_add {a : Cardinal.{u_1}} {b : Cardinal.{u_1}} {c : Cardinal.{u_1}} :
                a ^ (b + c) = a ^ b * a ^ c

                Porting note (#11229): Deprecated section. Remove.

                @[deprecated]
                theorem Cardinal.power_bit0 (a : Cardinal.{u_1}) (b : Cardinal.{u_1}) :
                a ^ bit0 b = a ^ b * a ^ b
                @[deprecated]
                theorem Cardinal.power_bit1 (a : Cardinal.{u_1}) (b : Cardinal.{u_1}) :
                a ^ bit1 b = a ^ b * a ^ b * a
                @[simp]
                theorem Cardinal.one_power {a : Cardinal.{u_1}} :
                1 ^ a = 1
                @[simp]
                theorem Cardinal.zero_power {a : Cardinal.{u_1}} :
                a 00 ^ a = 0
                theorem Cardinal.mul_power {a : Cardinal.{u_1}} {b : Cardinal.{u_1}} {c : Cardinal.{u_1}} :
                (a * b) ^ c = a ^ c * b ^ c
                theorem Cardinal.power_mul {a : Cardinal.{u_1}} {b : Cardinal.{u_1}} {c : Cardinal.{u_1}} :
                a ^ (b * c) = (a ^ b) ^ c
                @[simp]
                theorem Cardinal.pow_cast_right (a : Cardinal.{u}) (n : ) :
                a ^ n = a ^ n

                Porting note (#11229): Deprecated section. Remove.

                @[simp]
                theorem Cardinal.mk_set {α : Type u} :
                @[simp]
                theorem Cardinal.mk_powerset {α : Type u} (s : Set α) :

                A variant of Cardinal.mk_set expressed in terms of a Set instead of a Type.

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                • One or more equations did not get rendered due to their size.
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                • One or more equations did not get rendered due to their size.
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                • One or more equations did not get rendered due to their size.
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                • One or more equations did not get rendered due to their size.
                theorem Cardinal.power_le_power_left {a : Cardinal.{u_1}} {b : Cardinal.{u_1}} {c : Cardinal.{u_1}} :
                a 0b ca ^ b a ^ c
                theorem Cardinal.self_le_power (a : Cardinal.{u_1}) {b : Cardinal.{u_1}} (hb : 1 b) :
                a a ^ b
                theorem Cardinal.cantor (a : Cardinal.{u}) :
                a < 2 ^ a

                Cantor's theorem

                theorem Cardinal.power_pos {a : Cardinal.{u_1}} (b : Cardinal.{u_1}) (ha : 0 < a) :
                0 < a ^ b
                theorem Cardinal.lt_wf :
                WellFounded fun (x x_1 : Cardinal.{u}) => x < x_1
                instance Cardinal.wo :
                IsWellOrder Cardinal.{u} fun (x x_1 : Cardinal.{u}) => x < x_1
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                @[simp]
                theorem Cardinal.sInf_eq_zero_iff {s : Set Cardinal.{u_1}} :
                sInf s = 0 s = ∃ a ∈ s, a = 0
                theorem Cardinal.iInf_eq_zero_iff {ι : Sort u_1} {f : ιCardinal.{u_2}} :
                ⨅ (i : ι), f i = 0 IsEmpty ι ∃ (i : ι), f i = 0

                Note that the successor of c is not the same as c + 1 except in the case of finite c.

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                • One or more equations did not get rendered due to their size.

                A cardinal is a limit if it is not zero or a successor cardinal. Note that ℵ₀ is a limit cardinal by this definition, but 0 isn't.

                Use IsSuccLimit if you want to include the c = 0 case.

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                  The indexed sum of cardinals is the cardinality of the indexed disjoint union, i.e. sigma type.

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                    theorem Cardinal.le_sum {ι : Type u_1} (f : ιCardinal.{max u_1 u_2} ) (i : ι) :
                    @[simp]
                    theorem Cardinal.mk_sigma {ι : Type u_2} (f : ιType u_1) :
                    Cardinal.mk ((i : ι) × f i) = Cardinal.sum fun (i : ι) => Cardinal.mk (f i)
                    theorem Cardinal.sum_const' (ι : Type u) (a : Cardinal.{u}) :
                    (Cardinal.sum fun (x : ι) => a) = Cardinal.mk ι * a
                    @[simp]
                    theorem Cardinal.sum_add_distrib {ι : Type u_1} (f : ιCardinal.{u_2}) (g : ιCardinal.{u_2}) :
                    @[simp]
                    theorem Cardinal.sum_add_distrib' {ι : Type u_1} (f : ιCardinal.{u_2}) (g : ιCardinal.{u_2}) :
                    (Cardinal.sum fun (i : ι) => f i + g i) = Cardinal.sum f + Cardinal.sum g
                    theorem Cardinal.sum_le_sum {ι : Type u_1} (f : ιCardinal.{u_2}) (g : ιCardinal.{u_2}) (H : ∀ (i : ι), f i g i) :
                    theorem Cardinal.mk_le_mk_mul_of_mk_preimage_le {α : Type u} {β : Type u} {c : Cardinal.{u}} (f : αβ) (hf : ∀ (b : β), Cardinal.mk (f ⁻¹' {b}) c) :

                    The range of an indexed cardinal function, whose outputs live in a higher universe than the inputs, is always bounded above.

                    A set of cardinals is bounded above iff it's small, i.e. it corresponds to a usual ZFC set.

                    theorem Cardinal.iSup_of_empty {ι : Sort u_1} (f : ιCardinal.{u_2}) [IsEmpty ι] :
                    iSup f = 0

                    A variant of ciSup_of_empty but with 0 on the RHS for convenience

                    theorem Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit {ι : Type u} (f : ιCardinal.{v}) (ω : Cardinal.{v}) (hω : ¬Order.IsSuccLimit ω) (h : ⨆ (i : ι), f i = ω) :
                    ∃ (i : ι), f i = ω
                    theorem Cardinal.exists_eq_of_iSup_eq_of_not_isLimit {ι : Type u} [hι : Nonempty ι] (f : ιCardinal.{v}) (hf : BddAbove (Set.range f)) (ω : Cardinal.{v}) (hω : ¬Cardinal.IsLimit ω) (h : ⨆ (i : ι), f i = ω) :
                    ∃ (i : ι), f i = ω

                    The indexed product of cardinals is the cardinality of the Pi type (dependent product).

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                      @[simp]
                      theorem Cardinal.mk_pi {ι : Type u} (α : ιType v) :
                      Cardinal.mk ((i : ι) → α i) = Cardinal.prod fun (i : ι) => Cardinal.mk (α i)
                      theorem Cardinal.prod_const' (ι : Type u) (a : Cardinal.{u}) :
                      (Cardinal.prod fun (x : ι) => a) = a ^ Cardinal.mk ι
                      theorem Cardinal.prod_le_prod {ι : Type u_1} (f : ιCardinal.{u_2}) (g : ιCardinal.{u_2}) (H : ∀ (i : ι), f i g i) :
                      @[simp]
                      theorem Cardinal.prod_eq_zero {ι : Type u_1} (f : ιCardinal.{u}) :
                      Cardinal.prod f = 0 ∃ (i : ι), f i = 0
                      theorem Cardinal.prod_ne_zero {ι : Type u_1} (f : ιCardinal.{u_2}) :
                      Cardinal.prod f 0 ∀ (i : ι), f i 0
                      theorem Cardinal.prod_eq_of_fintype {α : Type u} [h : Fintype α] (f : αCardinal.{v}) :
                      Cardinal.prod f = Cardinal.lift.{u, v} (Finset.prod Finset.univ fun (i : α) => f i)
                      @[simp]
                      theorem Cardinal.lift_iInf {ι : Sort u_1} (f : ιCardinal.{v}) :

                      The lift of a supremum is the supremum of the lifts.

                      theorem Cardinal.lift_iSup {ι : Type v} {f : ιCardinal.{w}} (hf : BddAbove (Set.range f)) :

                      The lift of a supremum is the supremum of the lifts.

                      theorem Cardinal.lift_iSup_le {ι : Type v} {f : ιCardinal.{w}} {t : Cardinal.{max u w} } (hf : BddAbove (Set.range f)) (w : ∀ (i : ι), Cardinal.lift.{u, w} (f i) t) :

                      To prove that the lift of a supremum is bounded by some cardinal t, it suffices to show that the lift of each cardinal is bounded by t.

                      @[simp]
                      theorem Cardinal.lift_iSup_le_iff {ι : Type v} {f : ιCardinal.{w}} (hf : BddAbove (Set.range f)) {t : Cardinal.{max u w} } :
                      theorem Cardinal.lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ιCardinal.{w}} {f' : ι'Cardinal.{w'}} (hf : BddAbove (Set.range f)) (hf' : BddAbove (Set.range f')) {g : ιι'} (h : ∀ (i : ι), Cardinal.lift.{w', w} (f i) Cardinal.lift.{w, w'} (f' (g i))) :

                      To prove an inequality between the lifts to a common universe of two different supremums, it suffices to show that the lift of each cardinal from the smaller supremum if bounded by the lift of some cardinal from the larger supremum.

                      theorem Cardinal.lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ιCardinal.{v}} {f' : ι'Cardinal.{v'}} (hf : BddAbove (Set.range f)) (hf' : BddAbove (Set.range f')) (g : ιι') (h : ∀ (i : ι), Cardinal.lift.{v', v} (f i) Cardinal.lift.{v, v'} (f' (g i))) :

                      A variant of lift_iSup_le_lift_iSup with universes specialized via w = v and w' = v'. This is sometimes necessary to avoid universe unification issues.

                      ℵ₀ is the smallest infinite cardinal.

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                        ℵ₀ is the smallest infinite cardinal.

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                          Properties about the cast from #

                          theorem Cardinal.mk_fin (n : ) :
                          Cardinal.mk (Fin n) = n
                          @[simp]
                          @[simp]
                          @[simp]
                          @[simp]
                          @[simp]
                          theorem Cardinal.mk_coe_finset {α : Type u} {s : Finset α} :
                          Cardinal.mk { x : α // x s } = s.card
                          theorem Cardinal.card_le_of_finset {α : Type u_1} (s : Finset α) :
                          s.card Cardinal.mk α
                          theorem Cardinal.natCast_pow {m : } {n : } :
                          (m ^ n) = m ^ n
                          theorem Cardinal.natCast_le {m : } {n : } :
                          m n m n
                          theorem Cardinal.natCast_lt {m : } {n : } :
                          m < n m < n
                          theorem Cardinal.natCast_inj {m : } {n : } :
                          m = n m = n
                          @[simp]
                          theorem Cardinal.nat_succ (n : ) :
                          (Nat.succ n) = Order.succ n
                          theorem Cardinal.succ_natCast (n : ) :
                          Order.succ n = n + 1
                          theorem Cardinal.natCast_add_one_le_iff {n : } {c : Cardinal.{u_1}} :
                          n + 1 c n < c
                          theorem Cardinal.exists_finset_le_card (α : Type u_1) (n : ) (h : n Cardinal.mk α) :
                          ∃ (s : Finset α), n s.card
                          theorem Cardinal.card_le_of {α : Type u} {n : } (H : ∀ (s : Finset α), s.card n) :
                          theorem Cardinal.cantor' (a : Cardinal.{u_1}) {b : Cardinal.{u_1}} (hb : 1 < b) :
                          a < b ^ a
                          @[simp]
                          theorem Cardinal.lt_aleph0 {c : Cardinal.{u_1}} :
                          c < Cardinal.aleph0 ∃ (n : ), c = n
                          theorem Cardinal.exists_eq_natCast_of_iSup_eq {ι : Type u} [Nonempty ι] (f : ιCardinal.{v}) (hf : BddAbove (Set.range f)) (n : ) (h : ⨆ (i : ι), f i = n) :
                          ∃ (i : ι), f i = n
                          theorem Cardinal.mk_eq_nat_iff {α : Type u} {n : } :
                          Cardinal.mk α = n Nonempty (α Fin n)
                          theorem Set.Finite.lt_aleph0 {α : Type u} {S : Set α} :

                          Alias of the reverse direction of Cardinal.lt_aleph0_iff_set_finite.

                          @[simp]
                          theorem Cardinal.lt_aleph0_iff_subtype_finite {α : Type u} {p : αProp} :
                          Cardinal.mk { x : α // p x } < Cardinal.aleph0 Set.Finite {x : α | p x}

                          Alias of the reverse direction of Cardinal.le_aleph0_iff_set_countable.

                          @[simp]
                          theorem Cardinal.le_aleph0_iff_subtype_countable {α : Type u} {p : αProp} :
                          Cardinal.mk { x : α // p x } Cardinal.aleph0 Set.Countable {x : α | p x}
                          @[simp]
                          @[simp]
                          theorem Cardinal.exists_nat_eq_of_le_nat {c : Cardinal.{u_1}} {n : } (h : c n) :
                          ∃ m ≤ n, c = m
                          theorem Cardinal.sum_lt_prod {ι : Type u_1} (f : ιCardinal.{u_2}) (g : ιCardinal.{u_2}) (H : ∀ (i : ι), f i < g i) :

                          König's theorem

                          Cardinalities of sets: cardinality of empty, finite sets, unions, subsets etc.

                          theorem Cardinal.mk_singleton {α : Type u} (x : α) :
                          Cardinal.mk {x} = 1
                          @[simp]
                          theorem Cardinal.mk_vector (α : Type u) (n : ) :
                          theorem Cardinal.mk_quot_le {α : Type u} {r : ααProp} :
                          theorem Cardinal.mk_subtype_le_of_subset {α : Type u} {p : αProp} {q : αProp} (h : ∀ ⦃x : α⦄, p xq x) :
                          theorem Cardinal.mk_emptyCollection_iff {α : Type u} {s : Set α} :
                          Cardinal.mk s = 0 s =
                          @[simp]
                          theorem Cardinal.mk_univ {α : Type u} :
                          Cardinal.mk Set.univ = Cardinal.mk α
                          theorem Cardinal.mk_image_le {α : Type u} {β : Type u} {f : αβ} {s : Set α} :
                          theorem Cardinal.mk_image_le_lift {α : Type u} {β : Type v} {f : αβ} {s : Set α} :
                          theorem Cardinal.mk_range_le {α : Type u} {β : Type u} {f : αβ} :
                          theorem Cardinal.mk_range_eq {α : Type u} {β : Type u} (f : αβ) (h : Function.Injective f) :
                          theorem Cardinal.mk_image_eq_of_injOn {α : Type u} {β : Type u} (f : αβ) (s : Set α) (h : Set.InjOn f s) :
                          Cardinal.mk (f '' s) = Cardinal.mk s
                          theorem Cardinal.mk_image_eq_of_injOn_lift {α : Type u} {β : Type v} (f : αβ) (s : Set α) (h : Set.InjOn f s) :
                          theorem Cardinal.mk_image_eq {α : Type u} {β : Type u} {f : αβ} {s : Set α} (hf : Function.Injective f) :
                          Cardinal.mk (f '' s) = Cardinal.mk s
                          theorem Cardinal.mk_image_eq_lift {α : Type u} {β : Type v} (f : αβ) (s : Set α) (h : Function.Injective f) :
                          theorem Cardinal.mk_iUnion_le_sum_mk {α : Type u} {ι : Type u} {f : ιSet α} :
                          Cardinal.mk (⋃ (i : ι), f i) Cardinal.sum fun (i : ι) => Cardinal.mk (f i)
                          theorem Cardinal.mk_iUnion_le_sum_mk_lift {α : Type u} {ι : Type v} {f : ιSet α} :
                          Cardinal.lift.{v, u} (Cardinal.mk (⋃ (i : ι), f i)) Cardinal.sum fun (i : ι) => Cardinal.mk (f i)
                          theorem Cardinal.mk_iUnion_eq_sum_mk {α : Type u} {ι : Type u} {f : ιSet α} (h : Pairwise fun (i j : ι) => Disjoint (f i) (f j)) :
                          Cardinal.mk (⋃ (i : ι), f i) = Cardinal.sum fun (i : ι) => Cardinal.mk (f i)
                          theorem Cardinal.mk_iUnion_eq_sum_mk_lift {α : Type u} {ι : Type v} {f : ιSet α} (h : Pairwise fun (i j : ι) => Disjoint (f i) (f j)) :
                          Cardinal.lift.{v, u} (Cardinal.mk (⋃ (i : ι), f i)) = Cardinal.sum fun (i : ι) => Cardinal.mk (f i)
                          theorem Cardinal.mk_iUnion_le {α : Type u} {ι : Type u} (f : ιSet α) :
                          Cardinal.mk (⋃ (i : ι), f i) Cardinal.mk ι * ⨆ (i : ι), Cardinal.mk (f i)
                          theorem Cardinal.mk_iUnion_le_lift {α : Type u} {ι : Type v} (f : ιSet α) :
                          theorem Cardinal.mk_sUnion_le {α : Type u} (A : Set (Set α)) :
                          Cardinal.mk (⋃₀ A) Cardinal.mk A * ⨆ (s : A), Cardinal.mk s
                          theorem Cardinal.mk_biUnion_le {ι : Type u} {α : Type u} (A : ιSet α) (s : Set ι) :
                          Cardinal.mk (⋃ x ∈ s, A x) Cardinal.mk s * ⨆ (x : s), Cardinal.mk (A x)
                          theorem Cardinal.mk_biUnion_le_lift {α : Type u} {ι : Type v} (A : ιSet α) (s : Set ι) :
                          Cardinal.lift.{v, u} (Cardinal.mk (⋃ x ∈ s, A x)) Cardinal.lift.{u, v} (Cardinal.mk s) * ⨆ (x : s), Cardinal.lift.{v, u} (Cardinal.mk (A x))
                          theorem Cardinal.mk_set_eq_nat_iff_finset {α : Type u_1} {s : Set α} {n : } :
                          Cardinal.mk s = n ∃ (t : Finset α), t = s t.card = n
                          theorem Cardinal.mk_eq_nat_iff_finset {α : Type u} {n : } :
                          Cardinal.mk α = n ∃ (t : Finset α), t = Set.univ t.card = n
                          theorem Cardinal.mk_eq_nat_iff_fintype {α : Type u} {n : } :
                          Cardinal.mk α = n ∃ (h : Fintype α), Fintype.card α = n
                          theorem Cardinal.mk_union_add_mk_inter {α : Type u} {S : Set α} {T : Set α} :
                          Cardinal.mk (S T) + Cardinal.mk (S T) = Cardinal.mk S + Cardinal.mk T
                          theorem Cardinal.mk_union_le {α : Type u} (S : Set α) (T : Set α) :

                          The cardinality of a union is at most the sum of the cardinalities of the two sets.

                          theorem Cardinal.mk_union_of_disjoint {α : Type u} {S : Set α} {T : Set α} (H : Disjoint S T) :
                          theorem Cardinal.mk_insert {α : Type u} {s : Set α} {a : α} (h : as) :
                          Cardinal.mk (insert a s) = Cardinal.mk s + 1
                          theorem Cardinal.mk_insert_le {α : Type u} {s : Set α} {a : α} :
                          theorem Cardinal.mk_sum_compl {α : Type u_1} (s : Set α) :
                          theorem Cardinal.mk_le_mk_of_subset {α : Type u_1} {s : Set α} {t : Set α} (h : s t) :
                          theorem Cardinal.mk_le_iff_forall_finset_subset_card_le {α : Type u} {n : } {t : Set α} :
                          Cardinal.mk t n ∀ (s : Finset α), s ts.card n
                          theorem Cardinal.mk_subtype_mono {α : Type u} {p : αProp} {q : αProp} (h : ∀ (x : α), p xq x) :
                          Cardinal.mk { x : α // p x } Cardinal.mk { x : α // q x }
                          theorem Cardinal.le_mk_diff_add_mk {α : Type u} (S : Set α) (T : Set α) :
                          theorem Cardinal.mk_diff_add_mk {α : Type u} {S : Set α} {T : Set α} (h : T S) :
                          Cardinal.mk (S \ T) + Cardinal.mk T = Cardinal.mk S
                          theorem Cardinal.mk_subtype_of_equiv {α : Type u} {β : Type u} (p : βProp) (e : α β) :
                          Cardinal.mk { a : α // p (e a) } = Cardinal.mk { b : β // p b }
                          theorem Cardinal.mk_sep {α : Type u} (s : Set α) (t : αProp) :
                          Cardinal.mk {x : α | x s t x} = Cardinal.mk {x : s | t x}
                          theorem Cardinal.mk_preimage_of_injective_of_subset_range {α : Type u} {β : Type u} (f : αβ) (s : Set β) (h : Function.Injective f) (h2 : s Set.range f) :
                          theorem Cardinal.mk_preimage_of_injective {α : Type u} {β : Type u} (f : αβ) (s : Set β) (h : Function.Injective f) :
                          theorem Cardinal.mk_preimage_of_subset_range {α : Type u} {β : Type u} (f : αβ) (s : Set β) (h : s Set.range f) :
                          theorem Cardinal.mk_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : αβ) {s : Set α} {t : Set β} (h : t f '' s) :
                          theorem Cardinal.mk_subset_ge_of_subset_image {α : Type u} {β : Type u} (f : αβ) {s : Set α} {t : Set β} (h : t f '' s) :
                          Cardinal.mk t Cardinal.mk {x : α | x s f x t}
                          theorem Cardinal.le_mk_iff_exists_subset {c : Cardinal.{u}} {α : Type u} {s : Set α} :
                          c Cardinal.mk s ∃ p ⊆ s, Cardinal.mk p = c
                          theorem Cardinal.two_le_iff {α : Type u} :
                          2 Cardinal.mk α ∃ (x : α) (y : α), x y
                          theorem Cardinal.two_le_iff' {α : Type u} (x : α) :
                          2 Cardinal.mk α ∃ (y : α), y x
                          theorem Cardinal.mk_eq_two_iff {α : Type u} :
                          Cardinal.mk α = 2 ∃ (x : α) (y : α), x y {x, y} = Set.univ
                          theorem Cardinal.mk_eq_two_iff' {α : Type u} (x : α) :
                          Cardinal.mk α = 2 ∃! (y : α), y x
                          theorem Cardinal.exists_not_mem_of_length_lt {α : Type u_1} (l : List α) (h : (List.length l) < Cardinal.mk α) :
                          ∃ (z : α), zl
                          theorem Cardinal.three_le {α : Type u_1} (h : 3 Cardinal.mk α) (x : α) (y : α) :
                          ∃ (z : α), z x z y

                          The function a ^< b, defined as the supremum of a ^ c for c < b.

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                            The function a ^< b, defined as the supremum of a ^ c for c < b.

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                            • One or more equations did not get rendered due to their size.
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                              theorem Cardinal.le_powerlt {b : Cardinal.{u}} {c : Cardinal.{u}} (a : Cardinal.{u}) (h : c < b) :
                              a ^ c a ^< b
                              theorem Cardinal.powerlt_le {a : Cardinal.{u}} {b : Cardinal.{u}} {c : Cardinal.{u}} :
                              a ^< b c x < b, a ^ x c
                              theorem Cardinal.powerlt_min {a : Cardinal.{u_1}} {b : Cardinal.{u_1}} {c : Cardinal.{u_1}} :
                              a ^< min b c = min (a ^< b) (a ^< c)
                              theorem Cardinal.powerlt_max {a : Cardinal.{u_1}} {b : Cardinal.{u_1}} {c : Cardinal.{u_1}} :
                              a ^< max b c = max (a ^< b) (a ^< c)
                              theorem Cardinal.zero_powerlt {a : Cardinal.{u_1}} (h : a 0) :
                              0 ^< a = 1
                              @[simp]

                              The cardinality of a nontrivial module over a ring is at least the cardinality of the ring if there are no zero divisors (for instance if the ring is a field)