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

Theorem opcom 4452
Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
Hypotheses
Ref Expression
opcom.1  |-  A  e. 
_V
opcom.2  |-  B  e. 
_V
Assertion
Ref Expression
opcom  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  A  =  B )

Proof of Theorem opcom
StepHypRef Expression
1 opcom.1 . . 3  |-  A  e. 
_V
2 opcom.2 . . 3  |-  B  e. 
_V
31, 2opth 4437 . 2  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  ( A  =  B  /\  B  =  A )
)
4 eqcom 2440 . . 3  |-  ( B  =  A  <->  A  =  B )
54anbi2i 677 . 2  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
6 anidm 627 . 2  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
73, 5, 63bitri 264 1  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
  Copyright terms: Public domain W3C validator