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Theorem elpr 3613
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 ∈ V
Assertion
Ref Expression
elpr (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpr
StepHypRef Expression
1 elpr.1 . 2 𝐴 ∈ V
2 elprg 3612 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
31, 2ax-mp 5 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wb 105  wo 708   = wceq 1353  wcel 2148  Vcvv 2737  {cpr 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599
This theorem is referenced by:  prmg  3713  difprsnss  3730  preqr1  3768  preq12b  3770  prel12  3771  pwprss  3805  pwtpss  3806  unipr  3823  intpr  3876  zfpair2  4210  elop  4231  ordtri2or2exmidlem  4525  onsucelsucexmidlem  4528  en2lp  4553  reg3exmidlemwe  4578  xpsspw  4738  acexmidlem2  5871  2oconcl  6439  exmidpw  6907  exmidpweq  6908  renfdisj  8016  fzpr  10076  maxabslemval  11216  xrmaxiflemval  11257  isprm2  12116  bj-zfpair2  14632  ss1oel2o  14713
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