Theorem cnvcnvsn 4907

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4913, 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 4810	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 4810	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 2622	. . . 4 ⊢ 𝑦 ∈ V
4		vex 2622	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	opelcnv 4618	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩})
6		ancom 262	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴))
7	3, 4	opth 4064	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐵))
8	4, 3	opth 4064	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴))
9	6, 7, 8	3bitr4i 210	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
10	3, 4	opex 4056	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
11	10	elsn 3462	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
12	4, 3	opex 4056	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13	12	elsn 3462	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
14	9, 11, 13	3bitr4i 210	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
15	4, 3	opelcnv 4618	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 4618	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
17	14, 15, 16	3bitr4i 210	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
18	5, 17	bitri 182	. 2 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
19	1, 2, 18	eqrelriiv 4532	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 102   = wceq 1289   ∈ wcel 1438  {csn 3446  ⟨cop 3449  ^◡ccnv 4437

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446

This theorem is referenced by:  rnsnopg  4909  cnvsn  4913