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Theorem elpr2 3463
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 3461 . . 3  |-  ( A  e.  { B ,  C }  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
21ibi 174 . 2  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
3 elpr2.1 . . . . . 6  |-  B  e. 
_V
4 eleq1 2150 . . . . . 6  |-  ( A  =  B  ->  ( A  e.  _V  <->  B  e.  _V ) )
53, 4mpbiri 166 . . . . 5  |-  ( A  =  B  ->  A  e.  _V )
6 elpr2.2 . . . . . 6  |-  C  e. 
_V
7 eleq1 2150 . . . . . 6  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
86, 7mpbiri 166 . . . . 5  |-  ( A  =  C  ->  A  e.  _V )
95, 8jaoi 671 . . . 4  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  _V )
10 elprg 3461 . . . 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 175 . 2  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  { B ,  C } )
132, 12impbii 124 1  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   _Vcvv 2619   {cpr 3442
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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  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-sn 3447  df-pr 3448
This theorem is referenced by:  elxr  9216
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