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Theorem cnvcnvsn 4985
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4991, 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 4887 . 2 Rel {⟨𝐴, 𝐵⟩}
2 relcnv 4887 . 2 Rel {⟨𝐵, 𝐴⟩}
3 vex 2663 . . . 4 𝑦 ∈ V
4 vex 2663 . . . 4 𝑥 ∈ V
53, 4opelcnv 4691 . . 3 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
6 ancom 264 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐵) ↔ (𝑥 = 𝐵𝑦 = 𝐴))
73, 4opth 4129 . . . . . 6 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐵))
84, 3opth 4129 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐴))
96, 7, 83bitr4i 211 . . . . 5 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
103, 4opex 4121 . . . . . 6 𝑦, 𝑥⟩ ∈ V
1110elsn 3513 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
124, 3opex 4121 . . . . . 6 𝑥, 𝑦⟩ ∈ V
1312elsn 3513 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
149, 11, 133bitr4i 211 . . . 4 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
154, 3opelcnv 4691 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
163, 4opelcnv 4691 . . . 4 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
1714, 15, 163bitr4i 211 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
185, 17bitri 183 . 2 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
191, 2, 18eqrelriiv 4603 1 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wcel 1465  {csn 3497  cop 3500  ccnv 4508
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by:  rnsnopg  4987  cnvsn  4991
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