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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.
StepHypRef 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
53, 4opelcnv 4618 . . 3 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
6 ancom 262 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐵) ↔ (𝑥 = 𝐵𝑦 = 𝐴))
73, 4opth 4064 . . . . . 6 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐵))
84, 3opth 4064 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐴))
96, 7, 83bitr4i 210 . . . . 5 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
103, 4opex 4056 . . . . . 6 𝑦, 𝑥⟩ ∈ V
1110elsn 3462 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
124, 3opex 4056 . . . . . 6 𝑥, 𝑦⟩ ∈ V
1312elsn 3462 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
149, 11, 133bitr4i 210 . . . 4 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
154, 3opelcnv 4618 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
163, 4opelcnv 4618 . . . 4 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
1714, 15, 163bitr4i 210 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
185, 17bitri 182 . 2 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
191, 2, 18eqrelriiv 4532 1 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
Colors of variables: wff set class
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
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