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Theorem elpr2 3563
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 3561 . . 3  |-  ( A  e.  { B ,  C }  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
21ibi 234 . 2  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
3 elpr2.1 . . . . . 6  |-  B  e. 
_V
4 eleq1 2313 . . . . . 6  |-  ( A  =  B  ->  ( A  e.  _V  <->  B  e.  _V ) )
53, 4mpbiri 226 . . . . 5  |-  ( A  =  B  ->  A  e.  _V )
6 elpr2.2 . . . . . 6  |-  C  e. 
_V
7 eleq1 2313 . . . . . 6  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
86, 7mpbiri 226 . . . . 5  |-  ( A  =  C  ->  A  e.  _V )
95, 8jaoi 370 . . . 4  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  _V )
10 elprg 3561 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
119, 10syl 17 . . 3  |-  ( ( A  =  B  \/  A  =  C )  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C )
) )
1211ibir 235 . 2  |-  ( ( A  =  B  \/  A  =  C )  ->  A  e.  { B ,  C } )
132, 12impbii 182 1  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    = wceq 1619    e. wcel 1621   _Vcvv 2727   {cpr 3545
This theorem is referenced by:  elxr  10337  nofv  23478
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-sn 3550  df-pr 3551
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