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Theorem opcom 4068
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 4055 . 2  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  ( A  =  B  /\  B  =  A )
)
4 eqcom 2090 . . 3  |-  ( B  =  A  <->  A  =  B )
54anbi2i 445 . 2  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
6 anidm 388 . 2  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
73, 5, 63bitri 204 1  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   <.cop 3444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450
This theorem is referenced by: (None)
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