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Theorem elprg 3888
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 2444 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 eqeq1 2444 . . 3  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
31, 2orbi12d 709 . 2  |-  ( x  =  A  ->  (
( x  =  B  \/  x  =  C )  <->  ( A  =  B  \/  A  =  C ) ) )
4 dfpr2 3887 . 2  |-  { B ,  C }  =  {
x  |  ( x  =  B  \/  x  =  C ) }
53, 4elab2g 3103 1  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1369    e. wcel 1756   {cpr 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2969  df-un 3328  df-sn 3873  df-pr 3875
This theorem is referenced by:  eldifpr  3889  elpr  3890  elpr2  3891  elpri  3892  eltpg  3913  ifpr  3919  prid1g  3976  ordunpr  6432  hashtpg  12178  cnsubrg  17848  atandm  22246  nbgra0nb  23291  eupath2lem1  23549  eliccioo  26057
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