# Documentation

Mathlib.Data.Set.Opposite

# The opposite of a set #

The opposite of a set s is simply the set obtained by taking the opposite of each member of s.

def Set.op {α : Type u_1} (s : Set α) :

The opposite of a set s is the set obtained by taking the opposite of each member of s.

Equations
def Set.unop {α : Type u_1} (s : ) :
Set α

The unop of a set s is the set obtained by taking the unop of each member of s.

Equations
@[simp]
theorem Set.mem_op {α : Type u_1} {s : Set α} {a : αᵒᵖ} :
a
@[simp]
theorem Set.op_mem_op {α : Type u_1} {s : Set α} {a : α} :
a s
@[simp]
theorem Set.mem_unop {α : Type u_1} {s : } {a : α} :
a s
@[simp]
theorem Set.unop_mem_unop {α : Type u_1} {s : } {a : αᵒᵖ} :
a s
@[simp]
theorem Set.op_unop {α : Type u_1} (s : Set α) :
Set.unop () = s
@[simp]
theorem Set.unop_op {α : Type u_1} (s : ) :
Set.op () = s
@[simp]
theorem Set.opEquiv_self_symm_apply_coe {α : Type u_1} (s : Set α) (x : s) :
↑(↑() x) =
@[simp]
theorem Set.opEquiv_self_apply_coe {α : Type u_1} (s : Set α) (x : ↑()) :
↑(↑() x) =
def Set.opEquiv_self {α : Type u_1} (s : Set α) :
↑() s

The members of the opposite of a set are in bijection with the members of the set itself.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Set.opEquiv_apply {α : Type u_1} (s : Set α) :
Set.opEquiv s =
@[simp]
theorem Set.opEquiv_symm_apply {α : Type u_1} (s : ) :
↑(Equiv.symm Set.opEquiv) s =
def Set.opEquiv {α : Type u_1} :
Set α

Taking opposites as an equivalence of powersets.

Equations
• Set.opEquiv = { toFun := Set.op, invFun := Set.unop, left_inv := (_ : ∀ (s : Set α), Set.unop () = s), right_inv := (_ : ∀ (s : ), Set.op () = s) }
@[simp]
theorem Set.singleton_op {α : Type u_1} (x : α) :
Set.op {x} = {}
@[simp]
theorem Set.singleton_unop {α : Type u_1} (x : αᵒᵖ) :
Set.unop {x} = {}
@[simp]
theorem Set.singleton_op_unop {α : Type u_1} (x : α) :
= {x}
@[simp]
theorem Set.singleton_unop_op {α : Type u_1} (x : αᵒᵖ) :
= {x}