Theorem cnvsn 2636

Description: Converse of a singleton of an ordered pair.

Hypotheses

Ref	Expression
cnvsn.1	⊢ A ∈ V
cnvsn.2	⊢ B ∈ V

Assertion

Ref	Expression
cnvsn	⊢ ◡{⟨A, B⟩} = {⟨B, A⟩}

Proof of Theorem cnvsn

Step	Hyp	Ref	Expression
1		relcnv 2624	. 2 ⊢ Rel ◡{⟨A, B⟩}
2		cnvsn.2	. . 3 ⊢ B ∈ V
3	2	relsn 2485	. 2 ⊢ Rel {⟨B, A⟩}
4		ancom 333	. . 3 ⊢ ((y = B ∧ x = A) ↔ (x = A ∧ y = B))
5		opex 1893	. . . . 5 ⊢ ⟨y, x⟩ ∈ V
6	5	elsnc 1826	. . . 4 ⊢ (⟨y, x⟩ ∈ {⟨B, A⟩} ↔ ⟨y, x⟩ = ⟨B, A⟩)
7		visset 1350	. . . . 5 ⊢ y ∈ V
8		visset 1350	. . . . 5 ⊢ x ∈ V
9		cnvsn.1	. . . . 5 ⊢ A ∈ V
10	7, 8, 9	opth 1898	. . . 4 ⊢ (⟨y, x⟩ = ⟨B, A⟩ ↔ (y = B ∧ x = A))
11	6, 10	bitr 151	. . 3 ⊢ (⟨y, x⟩ ∈ {⟨B, A⟩} ↔ (y = B ∧ x = A))
12	7, 8	opelcnv 2518	. . . 4 ⊢ (⟨y, x⟩ ∈ ◡{⟨A, B⟩} ↔ ⟨x, y⟩ ∈ {⟨A, B⟩})
13		opex 1893	. . . . 5 ⊢ ⟨x, y⟩ ∈ V
14	13	elsnc 1826	. . . 4 ⊢ (⟨x, y⟩ ∈ {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
15	8, 7, 2	opth 1898	. . . 4 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
16	12, 14, 15	3bitr 155	. . 3 ⊢ (⟨y, x⟩ ∈ ◡{⟨A, B⟩} ↔ (x = A ∧ y = B))
17	4, 11, 16	3bitr4r 159	. 2 ⊢ (⟨y, x⟩ ∈ ◡{⟨A, B⟩} ↔ ⟨y, x⟩ ∈ {⟨B, A⟩})
18	1, 3, 17	cleqreli 2484	1 ⊢ ◡{⟨A, B⟩} = {⟨B, A⟩}

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ^◡ccnv 2409

This theorem is referenced by:  rnsnop 2637  op2ndb 2638  op2nda 2639  f1osn 2827  xpcomen 3343

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426

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