Theorem cnvcnvsn 5213

Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5219, 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 5114	. 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩}
2		relcnv 5114	. 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩}
3		vex 2805	. . . 4 ⊢ 𝑦 ∈ V
4		vex 2805	. . . 4 ⊢ 𝑥 ∈ V
5	3, 4	opelcnv 4912	. . 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 4329	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐵))
8	4, 3	opth 4329	. . . . . 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 4321	. . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
11	10	elsn 3685	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
12	4, 3	opex 4321	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13	12	elsn 3685	. . . . 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 4912	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
16	3, 4	opelcnv 4912	. . . 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 4820	1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩}

Syntax hints:   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {csn 3669  ⟨cop 3672  ^◡ccnv 4724

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733

This theorem is referenced by:  rnsnopg  5215  cnvsn  5219