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Theorem elpr 3618
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, 13-Sep-1995.)
Hypothesis
Ref Expression
elpr.1  |-  A  e. 
_V
Assertion
Ref Expression
elpr  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )

Proof of Theorem elpr
StepHypRef Expression
1 elpr.1 . 2  |-  A  e. 
_V
2 elprg 3617 . 2  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
31, 2ax-mp 10 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 2757   {cpr 3601
This theorem is referenced by:  difprsn  3716  preqr1  3746  preq12b  3748  prel12  3749  pwpr  3783  pwtp  3784  unipr  3801  intpr  3855  axpr  4171  zfpair2  4173  elop  4197  opthwiener  4226  xpsspw  4771  2oconcl  6456  pw2f1olem  6920  sdom2en01  7882  gruun  8382  renfdisj  8839  fzpr  10792  isprm2  12714  drngnidl  15929  indistopon  16686  dfcon2  17093  cnconn  17096  uncon  17103  txindis  17276  txcon  17331  filcon  17526  xpsdsval  17893  rolle  19285  dvivthlem1  19303  ang180lem3  20057  ang180lem4  20058  wilthlem2  20255  sqff1o  20368  ppiub  20391  perfectlem2  20417  lgslem1  20483  lgsdir2lem4  20513  lgsdir2lem5  20514  subfacp1lem1  23068  subfacp1lem4  23072  nosgnn0  23666  bpoly2  24153  bpoly3  24154  rankeq1o  24162  onsucconi  24237  cntrset  24955  fnckle  25398  pfsubkl  25400  abhp2  25528  divrngidl  26006  isfldidl  26046  wopprc  26476  pw2f1ocnv  26483  kelac2lem  26515  dihmeetlem2N  30640
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-un 3118  df-sn 3606  df-pr 3607
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