Documentation

def Set.Subsingleton {α : Type u} (s : Set α) :

Prop

A set s is a Subsingleton if it has at most one element.

Equations

s.Subsingleton = ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → x = y

Instances For

theorem Set.Subsingleton.anti {α : Type u} {s t : Set α} (ht : t.Subsingleton) (hst : s ⊆ t) :

theorem Set.Subsingleton.eq_singleton_of_mem {α : Type u} {s : Set α} (hs : s.Subsingleton) {x : α} (hx : x ∈ s) :

s = {x}

@[simp]

theorem Set.subsingleton_empty {α : Type u} :

∅.Subsingleton

@[simp]

theorem Set.subsingleton_singleton {α : Type u} {a : α} :

{a}.Subsingleton

theorem Set.subsingleton_of_subset_singleton {α : Type u} {a : α} {s : Set α} (h : s ⊆ {a}) :

theorem Set.subsingleton_of_forall_eq {α : Type u} {s : Set α} (a : α) (h : ∀ b ∈ s, b = a) :

theorem Set.subsingleton_iff_singleton {α : Type u} {s : Set α} {x : α} (hx : x ∈ s) :

s.Subsingleton ↔ s = {x}

theorem Set.Subsingleton.eq_empty_or_singleton {α : Type u} {s : Set α} (hs : s.Subsingleton) :

s = ∅ ∨ ∃ (x : α), s = {x}

theorem Set.Subsingleton.induction_on {α : Type u} {s : Set α} {p : Set α → Prop} (hs : s.Subsingleton) (he : p ∅) (h₁ : ∀ (x : α), p {x}) :

p s

theorem Set.subsingleton_univ {α : Type u} [Subsingleton α] :

univ.Subsingleton

theorem Set.subsingleton_of_univ_subsingleton {α : Type u} (h : univ.Subsingleton) :

Subsingleton α

@[simp]

theorem Set.subsingleton_univ_iff {α : Type u} :

univ.Subsingleton ↔ Subsingleton α

theorem Set.Subsingleton.inter_singleton {α : Type u} {a : α} {s : Set α} :

(s ∩ {a}).Subsingleton

theorem Set.Subsingleton.singleton_inter {α : Type u} {a : α} {s : Set α} :

({a} ∩ s).Subsingleton

theorem Set.subsingleton_of_subsingleton {α : Type u} [Subsingleton α] {s : Set α} :

theorem Set.subsingleton_isTop (α : Type u_1) [PartialOrder α] :

{x : α | IsTop x}.Subsingleton

theorem Set.subsingleton_isBot (α : Type u_1) [PartialOrder α] :

{x : α | IsBot x}.Subsingleton

theorem Set.exists_eq_singleton_iff_nonempty_subsingleton {α : Type u} {s : Set α} :

(∃ (a : α), s = {a}) ↔ s.Nonempty ∧ s.Subsingleton

@[simp]

theorem Set.subsingleton_coe {α : Type u} (s : Set α) :

Subsingleton ↑s ↔ s.Subsingleton

s, coerced to a type, is a subsingleton type if and only if s is a subsingleton set.

theorem Set.Subsingleton.coe_sort {α : Type u} {s : Set α} :

s.Subsingleton → Subsingleton ↑s

instance Set.subsingleton_coe_of_subsingleton {α : Type u} [Subsingleton α] {s : Set α} :

Subsingleton ↑s

The coe_sort of a set s in a subsingleton type is a subsingleton. For the corresponding result for Subtype, see subtype.subsingleton.

Nontrivial #

def Set.Nontrivial {α : Type u} (s : Set α) :

Prop

A set s is Set.Nontrivial if it has at least two distinct elements.

Equations

s.Nontrivial = ∃ x ∈ s, ∃ y ∈ s, x ≠ y

Instances For

theorem Set.nontrivial_of_mem_mem_ne {α : Type u} {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y) :

noncomputable def Set.Nontrivial.choose {α : Type u} {s : Set α} (hs : s.Nontrivial) :

α × α

Extract witnesses from s.nontrivial. This function might be used instead of case analysis on the argument. Note that it makes a proof depend on the classical.choice axiom.

Equations

hs.choose = (Exists.choose hs, ⋯.choose)

Instances For

theorem Set.Nontrivial.choose_fst_mem {α : Type u} {s : Set α} (hs : s.Nontrivial) :

hs.choose.fst ∈ s

theorem Set.Nontrivial.choose_snd_mem {α : Type u} {s : Set α} (hs : s.Nontrivial) :

hs.choose.snd ∈ s

theorem Set.Nontrivial.choose_fst_ne_choose_snd {α : Type u} {s : Set α} (hs : s.Nontrivial) :

hs.choose.fst ≠ hs.choose.snd

theorem Set.Nontrivial.mono {α : Type u} {s t : Set α} (hs : s.Nontrivial) (hst : s ⊆ t) :

t.Nontrivial

theorem Set.nontrivial_pair {α : Type u} {x y : α} (hxy : x ≠ y) :

{x, y}.Nontrivial

theorem Set.nontrivial_of_pair_subset {α : Type u} {s : Set α} {x y : α} (hxy : x ≠ y) (h : {x, y} ⊆ s) :

theorem Set.Nontrivial.pair_subset {α : Type u} {s : Set α} (hs : s.Nontrivial) :

∃ (x : α) (y : α), x ≠ y ∧ {x, y} ⊆ s

theorem Set.nontrivial_iff_pair_subset {α : Type u} {s : Set α} :

s.Nontrivial ↔ ∃ (x : α) (y : α), x ≠ y ∧ {x, y} ⊆ s

theorem Set.nontrivial_of_exists_ne {α : Type u} {s : Set α} {x : α} (hx : x ∈ s) (h : ∃ y ∈ s, y ≠ x) :

theorem Set.Nontrivial.exists_ne {α : Type u} {s : Set α} (hs : s.Nontrivial) (z : α) :

∃ x ∈ s, x ≠ z

theorem Set.nontrivial_iff_exists_ne {α : Type u} {s : Set α} {x : α} (hx : x ∈ s) :

s.Nontrivial ↔ ∃ y ∈ s, y ≠ x

theorem Set.nontrivial_of_lt {α : Type u} {s : Set α} [Preorder α] {x y : α} (hx : x ∈ s) (hy : y ∈ s) (hxy : x < y) :

theorem Set.nontrivial_of_exists_lt {α : Type u} {s : Set α} [Preorder α] (H : ∃ x ∈ s, ∃ y ∈ s, x < y) :

theorem Set.Nontrivial.exists_lt {α : Type u} {s : Set α} [LinearOrder α] (hs : s.Nontrivial) :

∃ x ∈ s, ∃ y ∈ s, x < y

theorem Set.nontrivial_iff_exists_lt {α : Type u} {s : Set α} [LinearOrder α] :

s.Nontrivial ↔ ∃ x ∈ s, ∃ y ∈ s, x < y

theorem Set.Nontrivial.nonempty {α : Type u} {s : Set α} (hs : s.Nontrivial) :

s.Nonempty

theorem Set.Nontrivial.ne_empty {α : Type u} {s : Set α} (hs : s.Nontrivial) :

s ≠ ∅

theorem Set.Nontrivial.not_subset_empty {α : Type u} {s : Set α} (hs : s.Nontrivial) :

¬s ⊆ ∅

@[simp]

theorem Set.not_nontrivial_empty {α : Type u} :

¬∅.Nontrivial

@[simp]

theorem Set.not_nontrivial_singleton {α : Type u} {x : α} :

¬{x}.Nontrivial

theorem Set.Nontrivial.ne_singleton {α : Type u} {s : Set α} {x : α} (hs : s.Nontrivial) :

s ≠ {x}

theorem Set.Nontrivial.not_subset_singleton {α : Type u} {s : Set α} {x : α} (hs : s.Nontrivial) :

¬s ⊆ {x}

theorem Set.nontrivial_univ {α : Type u} [Nontrivial α] :

univ.Nontrivial

theorem Set.nontrivial_of_univ_nontrivial {α : Type u} (h : univ.Nontrivial) :

Nontrivial α

@[simp]

theorem Set.nontrivial_univ_iff {α : Type u} :

univ.Nontrivial ↔ Nontrivial α

@[simp]

theorem Set.singleton_ne_univ {α : Type u} [Nontrivial α] (a : α) :

{a} ≠ univ

@[simp]

theorem Set.singleton_ssubset_univ {α : Type u} [Nontrivial α] (a : α) :

{a} ⊂ univ

theorem Set.nontrivial_of_nontrivial {α : Type u} {s : Set α} (hs : s.Nontrivial) :

Nontrivial α

@[simp]

theorem Set.nontrivial_coe_sort {α : Type u} {s : Set α} :

Nontrivial ↑s ↔ s.Nontrivial

s, coerced to a type, is a nontrivial type if and only if s is a nontrivial set.

theorem Set.Nontrivial.coe_sort {α : Type u} {s : Set α} :

s.Nontrivial → Nontrivial ↑s

Alias of the reverse direction of Set.nontrivial_coe_sort.

s, coerced to a type, is a nontrivial type if and only if s is a nontrivial set.

theorem Set.nontrivial_of_nontrivial_coe {α : Type u} {s : Set α} (hs : Nontrivial ↑s) :

Nontrivial α

A type with a set s whose coe_sort is a nontrivial type is nontrivial. For the corresponding result for Subtype, see Subtype.nontrivial_iff_exists_ne.

theorem Set.nontrivial_mono {α : Type u_1} {s t : Set α} (hst : s ⊆ t) (hs : Nontrivial ↑s) :

Nontrivial ↑t

@[simp]

theorem Set.not_subsingleton_iff {α : Type u} {s : Set α} :

¬s.Subsingleton ↔ s.Nontrivial

@[simp]

theorem Set.not_nontrivial_iff {α : Type u} {s : Set α} :

¬s.Nontrivial ↔ s.Subsingleton

theorem Set.Subsingleton.not_nontrivial {α : Type u} {s : Set α} :

s.Subsingleton → ¬s.Nontrivial

Alias of the reverse direction of Set.not_nontrivial_iff.

theorem Set.Nontrivial.not_subsingleton {α : Type u} {s : Set α} :

s.Nontrivial → ¬s.Subsingleton

Alias of the reverse direction of Set.not_subsingleton_iff.

theorem Set.subsingleton_or_nontrivial {α : Type u} (s : Set α) :

s.Subsingleton ∨ s.Nontrivial

theorem Set.eq_singleton_or_nontrivial {α : Type u} {a : α} {s : Set α} (ha : a ∈ s) :

s = {a} ∨ s.Nontrivial

theorem Set.nontrivial_iff_ne_singleton {α : Type u} {a : α} {s : Set α} (ha : a ∈ s) :

s.Nontrivial ↔ s ≠ {a}

theorem Set.Nonempty.exists_eq_singleton_or_nontrivial {α : Type u} {s : Set α} :

s.Nonempty → (∃ (a : α), s = {a}) ∨ s.Nontrivial

theorem Set.univ_eq_true_false :

univ = {True , False }

Monotonicity on singletons #

theorem Set.Subsingleton.monotoneOn {α : Type u} {β : Type v} {s : Set α} [Preorder α] [Preorder β] (f : α → β) (h : s.Subsingleton) :

MonotoneOn f s

theorem Set.Subsingleton.antitoneOn {α : Type u} {β : Type v} {s : Set α} [Preorder α] [Preorder β] (f : α → β) (h : s.Subsingleton) :

AntitoneOn f s

theorem Set.Subsingleton.strictMonoOn {α : Type u} {β : Type v} {s : Set α} [Preorder α] [Preorder β] (f : α → β) (h : s.Subsingleton) :

StrictMonoOn f s

theorem Set.Subsingleton.strictAntiOn {α : Type u} {β : Type v} {s : Set α} [Preorder α] [Preorder β] (f : α → β) (h : s.Subsingleton) :

StrictAntiOn f s

@[simp]

theorem Set.monotoneOn_singleton {α : Type u} {β : Type v} {a : α} [Preorder α] [Preorder β] (f : α → β) :

MonotoneOn f {a}

@[simp]

theorem Set.antitoneOn_singleton {α : Type u} {β : Type v} {a : α} [Preorder α] [Preorder β] (f : α → β) :

AntitoneOn f {a}

@[simp]

theorem Set.strictMonoOn_singleton {α : Type u} {β : Type v} {a : α} [Preorder α] [Preorder β] (f : α → β) :

StrictMonoOn f {a}