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

Theorem opcom 4260
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 4245 . 2  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  ( A  =  B  /\  B  =  A )
)
4 eqcom 2285 . . 3  |-  ( B  =  A  <->  A  =  B )
54anbi2i 675 . 2  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
6 anidm 625 . 2  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
73, 5, 63bitri 262 1  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
  Copyright terms: Public domain W3C validator