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Theorem elprg 3461
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
Assertion
Ref Expression
elprg  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )

Proof of Theorem elprg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2094 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 eqeq1 2094 . . 3  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
31, 2orbi12d 742 . 2  |-  ( x  =  A  ->  (
( x  =  B  \/  x  =  C )  <->  ( A  =  B  \/  A  =  C ) ) )
4 dfpr2 3460 . 2  |-  { B ,  C }  =  {
x  |  ( x  =  B  \/  x  =  C ) }
53, 4elab2g 2760 1  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   {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:  elpr  3462  elpr2  3463  elpri  3464  eltpg  3483  prid1g  3541  preqr1g  3605  m1expeven  9967  maxclpr  10620  minmax  10625
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