Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opcom Structured version   Visualization version   Unicode version

Theorem opcom 4695
 Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
Hypotheses
Ref Expression
opcom.1
opcom.2
Assertion
Ref Expression
opcom

Proof of Theorem opcom
StepHypRef Expression
1 opcom.1 . . 3
2 opcom.2 . . 3
31, 2opth 4676 . 2
4 eqcom 2458 . . 3
54anbi2i 700 . 2
6 anidm 650 . 2
73, 5, 63bitri 275 1
 Colors of variables: wff setvar class Syntax hints:   wb 188   wa 371   wceq 1444   wcel 1887  cvv 3045  cop 3974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator