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Theorem opcom 4590
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 4571 . 2  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  ( A  =  B  /\  B  =  A )
)
4 eqcom 2445 . . 3  |-  ( B  =  A  <->  A  =  B )
54anbi2i 694 . 2  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
6 anidm 644 . 2  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
73, 5, 63bitri 271 1  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977   <.cop 3888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889
This theorem is referenced by: (None)
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