Theorem cnvcnvsn 5205

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5211, 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 5106	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 5106	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 2802	. . . 4 ⊢ 𝑦 ∈ V
4		vex 2802	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	opelcnv 4904	. . 3 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩})
6		ancom 266	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴))
7	3, 4	opth 4323	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐵))
8	4, 3	opth 4323	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴))
9	6, 7, 8	3bitr4i 212	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
10	3, 4	opex 4315	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
11	10	elsn 3682	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
12	4, 3	opex 4315	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13	12	elsn 3682	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
14	9, 11, 13	3bitr4i 212	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
15	4, 3	opelcnv 4904	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 4904	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
17	14, 15, 16	3bitr4i 212	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
18	5, 17	bitri 184	. 2 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐵, 𝐴⟩})
19	1, 2, 18	eqrelriiv 4813	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {csn 3666  ⟨cop 3669  ^◡ccnv 4718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727

This theorem is referenced by:  rnsnopg  5207  cnvsn  5211