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Theorem opcom 4015
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 4002 . 2  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  ( A  =  B  /\  B  =  A )
)
4 eqcom 2058 . . 3  |-  ( B  =  A  <->  A  =  B )
54anbi2i 438 . 2  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
6 anidm 382 . 2  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
73, 5, 63bitri 199 1  |-  ( <. A ,  B >.  = 
<. B ,  A >.  <->  A  =  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   <.cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by: (None)
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