93 lines
3.0 KiB
Plaintext
93 lines
3.0 KiB
Plaintext
|
import Mathlib.Data.Set.Basic
|
|||
|
|
|
|||
|
|
import Common.Logic.Basic
|
|||
|
|
import Common.Set.Basic
|
|||
|
|
|
|||
|
|
namespace Set
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
Kazimierz Kuratowski's definition of an ordered pair.
|
|||
|
|
-/
|
|||
|
|
def OrderedPair (x y : α) : Set (Set α) := {{x}, {x, y}}
|
|||
|
|
|
|||
|
|
namespace OrderedPair
|
|||
|
|
|
|||
|
|
theorem ext_iff {x y u v : α}
|
|||
|
|
: (OrderedPair x y = OrderedPair u v) ↔ (x = u ∧ y = v) := by
|
|||
|
|
unfold OrderedPair
|
|||
|
|
apply Iff.intro
|
|||
|
|
· intro h
|
|||
|
|
have h' := h
|
|||
|
|
rw [Set.ext_iff] at h'
|
|||
|
|
have hu := h' {u}
|
|||
|
|
have huv := h' {u, v}
|
|||
|
|
simp only [mem_singleton_iff, mem_insert_iff, true_or, iff_true] at hu
|
|||
|
|
simp only [mem_singleton_iff, mem_insert_iff, or_true, iff_true] at huv
|
|||
|
|
apply Or.elim hu
|
|||
|
|
· apply Or.elim huv
|
|||
|
|
· -- #### Case 1
|
|||
|
|
-- `{u} = {x}` and `{u, v} = {x}`.
|
|||
|
|
intro huv_x hu_x
|
|||
|
|
rw [singleton_eq_singleton_iff] at hu_x
|
|||
|
|
rw [hu_x] at huv_x
|
|||
|
|
have hx_v := pair_eq_singleton_mem_imp_eq_self huv_x
|
|||
|
|
rw [hu_x, hx_v] at h
|
|||
|
|
simp only [mem_singleton_iff, insert_eq_of_mem] at h
|
|||
|
|
have := pair_eq_singleton_mem_imp_eq_self $
|
|||
|
|
pair_eq_singleton_mem_imp_eq_self h
|
|||
|
|
rw [← hx_v] at this
|
|||
|
|
exact ⟨hu_x.symm, this⟩
|
|||
|
|
· -- #### Case 2
|
|||
|
|
-- `{u} = {x}` and `{u, v} = {x, y}`.
|
|||
|
|
intro huv_xy hu_x
|
|||
|
|
rw [singleton_eq_singleton_iff] at hu_x
|
|||
|
|
rw [hu_x] at huv_xy
|
|||
|
|
by_cases hx_v : x = v
|
|||
|
|
· rw [hx_v] at huv_xy
|
|||
|
|
simp at huv_xy
|
|||
|
|
have := pair_eq_singleton_mem_imp_eq_self huv_xy.symm
|
|||
|
|
exact ⟨hu_x.symm, this⟩
|
|||
|
|
· rw [Set.ext_iff] at huv_xy
|
|||
|
|
have := huv_xy v
|
|||
|
|
simp at this
|
|||
|
|
apply Or.elim this
|
|||
|
|
· intro hv_x
|
|||
|
|
rw [hu_x, ← hv_x] at h
|
|||
|
|
simp at h
|
|||
|
|
have := pair_eq_singleton_mem_imp_eq_self $
|
|||
|
|
pair_eq_singleton_mem_imp_eq_self h
|
|||
|
|
exact ⟨hu_x.symm, this⟩
|
|||
|
|
· intro hv_y
|
|||
|
|
exact ⟨hu_x.symm, hv_y.symm⟩
|
|||
|
|
· apply Or.elim huv
|
|||
|
|
· -- #### Case 3
|
|||
|
|
-- `{u} = {x, y}` and `{u, v} = {x}`.
|
|||
|
|
intro huv_x hu_xy
|
|||
|
|
rw [Set.ext_iff] at huv_x
|
|||
|
|
have hu_x := huv_x u
|
|||
|
|
have hv_x := huv_x v
|
|||
|
|
simp only [mem_singleton_iff, mem_insert_iff, true_or, true_iff] at hu_x
|
|||
|
|
simp only [mem_singleton_iff, mem_insert_iff, or_true, true_iff] at hv_x
|
|||
|
|
rw [← hu_x] at hu_xy
|
|||
|
|
have := pair_eq_singleton_mem_imp_eq_self hu_xy.symm
|
|||
|
|
rw [hu_x, ← hv_x] at this
|
|||
|
|
exact ⟨hu_x.symm, this⟩
|
|||
|
|
· -- #### Case 4
|
|||
|
|
-- `{u} = {x, y}` and `{u, v} = {x, y}`.
|
|||
|
|
intro huv_xy hu_xy
|
|||
|
|
rw [Set.ext_iff] at hu_xy
|
|||
|
|
have hx_u := hu_xy x
|
|||
|
|
have hy_u := hu_xy y
|
|||
|
|
simp only [mem_singleton_iff, mem_insert_iff, true_or, iff_true] at hx_u
|
|||
|
|
simp only [mem_singleton_iff, mem_insert_iff, or_true, iff_true] at hy_u
|
|||
|
|
rw [hx_u, hy_u] at huv_xy
|
|||
|
|
simp only [mem_singleton_iff, insert_eq_of_mem] at huv_xy
|
|||
|
|
have := pair_eq_singleton_mem_imp_eq_self huv_xy
|
|||
|
|
rw [← this] at hy_u
|
|||
|
|
exact ⟨hx_u, hy_u⟩
|
|||
|
|
· intro h
|
|||
|
|
rw [h.left, h.right]
|
|||
|
|
|
|||
|
|
end OrderedPair
|
|||
|
|
|
|||
|
|
end Set
|