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Theorem elpr2 3517
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
Hypotheses
Ref Expression
elpr2.1  |-  B  e. 
_V
elpr2.2  |-  C  e. 
_V
Assertion
Ref Expression
elpr2  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )

Proof of Theorem elpr2
StepHypRef Expression
1 elprg 3515 . . 3  |-  ( A  e.  { B ,  C }  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
21ibi 175 . 2  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
3 elpr2.1 . . . . . 6  |-  B  e. 
_V
4 eleq1 2178 . . . . . 6  |-  ( A  =  B  ->  ( A  e.  _V  <->  B  e.  _V ) )
53, 4mpbiri 167 . . . . 5  |-  ( A  =  B  ->  A  e.  _V )
6 elpr2.2 . . . . . 6  |-  C  e. 
_V
7 eleq1 2178 . . . . . 6  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
86, 7mpbiri 167 . . . . 5  |-  ( A  =  C  ->  A  e.  _V )
95, 8jaoi 688 . . . 4  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  _V )
10 elprg 3515 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
119, 10syl 14 . . 3  |-  ( ( A  =  B  \/  A  =  C )  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
) )
1211ibir 176 . 2  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  { B ,  C } )
132, 12impbii 125 1  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   _Vcvv 2658   {cpr 3496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502
This theorem is referenced by:  elxr  9514
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