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Theorem opcom 4012
Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
Hypotheses
Ref Expression
opcom.1 𝐴 ∈ V
opcom.2 𝐵 ∈ V
Assertion
Ref Expression
opcom (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)

Proof of Theorem opcom
StepHypRef Expression
1 opcom.1 . . 3 𝐴 ∈ V
2 opcom.2 . . 3 𝐵 ∈ V
31, 2opth 3999 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝐴 = 𝐵𝐵 = 𝐴))
4 eqcom 2056 . . 3 (𝐵 = 𝐴𝐴 = 𝐵)
54anbi2i 438 . 2 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
6 anidm 382 . 2 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
73, 5, 63bitri 199 1 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1257  wcel 1407  Vcvv 2572  cop 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409
This theorem is referenced by: (None)
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