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Theorem elprg 4032
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 2458 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 eqeq1 2458 . . 3  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
31, 2orbi12d 707 . 2  |-  ( x  =  A  ->  (
( x  =  B  \/  x  =  C )  <->  ( A  =  B  \/  A  =  C ) ) )
4 dfpr2 4031 . 2  |-  { B ,  C }  =  {
x  |  ( x  =  B  \/  x  =  C ) }
53, 4elab2g 3245 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 366    = wceq 1398    e. wcel 1823   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019
This theorem is referenced by:  eldifpr  4033  elpr  4034  elpr2  4035  elpri  4036  eltpg  4058  ifpr  4064  prid1g  4122  ordunpr  6634  hashtpg  12507  cnsubrg  18673  atandm  23404  nbgra0nb  24631  eupath2lem1  25179  eliccioo  27861
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