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Theorem cnvcnvsn 4797
 Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4803, 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 4703 . 2 Rel {⟨𝐴, 𝐵⟩}
2 relcnv 4703 . 2 Rel {⟨𝐵, 𝐴⟩}
3 vex 2560 . . . 4 𝑦 ∈ V
4 vex 2560 . . . 4 𝑥 ∈ V
53, 4opelcnv 4517 . . 3 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
6 ancom 253 . . . . . 6 ((𝑦 = 𝐴𝑥 = 𝐵) ↔ (𝑥 = 𝐵𝑦 = 𝐴))
73, 4opth 3974 . . . . . 6 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐵))
84, 3opth 3974 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐴))
96, 7, 83bitr4i 201 . . . . 5 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
103, 4opex 3966 . . . . . 6 𝑦, 𝑥⟩ ∈ V
1110elsn 3391 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
124, 3opex 3966 . . . . . 6 𝑥, 𝑦⟩ ∈ V
1312elsn 3391 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐴⟩)
149, 11, 133bitr4i 201 . . . 4 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
154, 3opelcnv 4517 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
163, 4opelcnv 4517 . . . 4 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
1714, 15, 163bitr4i 201 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
185, 17bitri 173 . 2 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
191, 2, 18eqrelriiv 4434 1 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {csn 3375  ⟨cop 3378  ◡ccnv 4344 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353 This theorem is referenced by:  rnsnopg  4799  cnvsn  4803
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