Theorem cnvcnvsn 5530

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5536, 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 5422	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 5422	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
4		vex 3176	. . . 4 ⊢ 𝑦 ∈ V
5	3, 4	opelcnv 5226	. . 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 4871	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
8	4, 3	opth 4871	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴))
9	6, 7, 8	3bitr4i 291	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
10		opex 4859	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
11	10	elsn 4140	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
12		opex 4859	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
13	12	elsn 4140	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
14	9, 11, 13	3bitr4i 291	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
15	4, 3	opelcnv 5226	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 5226	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
17	14, 15, 16	3bitr4i 291	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
18	5, 17	bitri 263	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
19	1, 2, 18	eqrelriiv 5137	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  ⟨cop 4131  ^◡ccnv 5037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046

This theorem is referenced by:  rnsnopg  5532  cnvsn  5536  strlemor1  15796