Theorem cnvcnvsn 6212

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 6219, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem cnvcnvsn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6097	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 6097	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 3472	. . . 4 ⊢ 𝑥 ∈ V
4		vex 3472	. . . 4 ⊢ 𝑦 ∈ V
5	3, 4	opelcnv 5875	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩})
6		ancom 460	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴))
7	3, 4	opth 5469	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
8	4, 3	opth 5469	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴))
9	6, 7, 8	3bitr4i 303	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
10		opex 5457	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
11	10	elsn 4638	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
12		opex 5457	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
13	12	elsn 4638	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
14	9, 11, 13	3bitr4i 303	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
15	4, 3	opelcnv 5875	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 5875	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
17	14, 15, 16	3bitr4i 303	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
18	5, 17	bitri 275	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
19	1, 2, 18	eqrelriiv 5783	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {csn 4623  ⟨cop 4629  ^◡ccnv 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677

This theorem is referenced by:  rnsnopg  6214  cnvsng  6216