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Theorem opcom 3957
 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 3944 . 2 (⟨A, B⟩ = ⟨B, A⟩ ↔ (A = B B = A))
4 eqcom 2020 . . 3 (B = AA = B)
54anbi2i 433 . 2 ((A = B B = A) ↔ (A = B A = B))
6 anidm 376 . 2 ((A = B A = B) ↔ A = B)
73, 5, 63bitri 195 1 (⟨A, B⟩ = ⟨B, A⟩ ↔ A = B)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1226   ∈ wcel 1370  Vcvv 2531  ⟨cop 3349 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355 This theorem is referenced by: (None)
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