Theorem cnvcnvsn 6224

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 6231, 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 6110	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 6110	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 3466	. . . 4 ⊢ 𝑥 ∈ V
4		vex 3466	. . . 4 ⊢ 𝑦 ∈ V
5	3, 4	opelcnv 5871	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩})
6		ancom 465	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴))
7	3, 4	opth 5462	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
8	4, 3	opth 5462	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴))
9	6, 7, 8	3bitr4i 306	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
10		opex 5449	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
11	10	elsn 4609	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
12		opex 5449	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
13	12	elsn 4609	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
14	9, 11, 13	3bitr4i 306	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
15	4, 3	opelcnv 5871	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 5871	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
17	14, 15, 16	3bitr4i 306	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
18	5, 17	bitri 278	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
19	1, 2, 18	eqrelriiv 5780	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 400   = wceq 1568   ∈ wcel 2150  {csn 4594  ⟨cop 4600  ^◡ccnv 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673

This theorem is referenced by:  rnsnopg  6226  cnvsng  6228