Theorem cnvcnvsn 6078

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 6085, 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 5969	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 5969	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 3499	. . . 4 ⊢ 𝑥 ∈ V
4		vex 3499	. . . 4 ⊢ 𝑦 ∈ V
5	3, 4	opelcnv 5754	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩})
6		ancom 463	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴))
7	3, 4	opth 5370	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
8	4, 3	opth 5370	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴))
9	6, 7, 8	3bitr4i 305	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
10		opex 5358	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
11	10	elsn 4584	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
12		opex 5358	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
13	12	elsn 4584	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
14	9, 11, 13	3bitr4i 305	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
15	4, 3	opelcnv 5754	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 5754	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
17	14, 15, 16	3bitr4i 305	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
18	5, 17	bitri 277	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
19	1, 2, 18	eqrelriiv 5665	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {csn 4569  ⟨cop 4575  ^◡ccnv 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565

This theorem is referenced by:  rnsnopg  6080  cnvsng  6082