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Theorem opcom 5500
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 5475 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝐴 = 𝐵𝐵 = 𝐴))
4 eqcom 2739 . . 3 (𝐵 = 𝐴𝐴 = 𝐵)
54anbi2i 623 . 2 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
6 anidm 565 . 2 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
73, 5, 63bitri 296 1 (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cop 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634
This theorem is referenced by: (None)
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