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Theorem elpr 3512
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 3511 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
31, 2ax-mp 7 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 680   = wceq 1312  wcel 1461  Vcvv 2655  {cpr 3492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-sn 3497  df-pr 3498
This theorem is referenced by:  prmg  3608  difprsnss  3622  preqr1  3659  preq12b  3661  prel12  3662  pwprss  3696  pwtpss  3697  unipr  3714  intpr  3767  zfpair2  4090  elop  4111  ordtri2or2exmidlem  4399  onsucelsucexmidlem  4402  en2lp  4427  reg3exmidlemwe  4451  xpsspw  4609  acexmidlem2  5723  2oconcl  6288  exmidpw  6753  renfdisj  7742  fzpr  9744  maxabslemval  10866  xrmaxiflemval  10905  isprm2  11638  bj-zfpair2  12791  ss1oel2o  12872
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