Theorem cnvcnvsn 5023

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5029, 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 4925	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 4925	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 2692	. . . 4 ⊢ 𝑦 ∈ V
4		vex 2692	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	opelcnv 4729	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩})
6		ancom 264	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴))
7	3, 4	opth 4167	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐵))
8	4, 3	opth 4167	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴))
9	6, 7, 8	3bitr4i 211	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
10	3, 4	opex 4159	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
11	10	elsn 3548	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
12	4, 3	opex 4159	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13	12	elsn 3548	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
14	9, 11, 13	3bitr4i 211	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
15	4, 3	opelcnv 4729	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 4729	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
17	14, 15, 16	3bitr4i 211	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
18	5, 17	bitri 183	. 2 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
19	1, 2, 18	eqrelriiv 4641	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 103   = wceq 1332   ∈ wcel 1481  {csn 3532  ⟨cop 3535  ^◡ccnv 4546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555

This theorem is referenced by:  rnsnopg  5025  cnvsn  5029