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Theorem elpr 3443
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 3442 . 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 103  wo 662   = wceq 1285  wcel 1434  Vcvv 2612  {cpr 3423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429
This theorem is referenced by:  prmg  3535  difprsnss  3549  preqr1  3586  preq12b  3588  prel12  3589  pwprss  3623  pwtpss  3624  unipr  3641  intpr  3694  zfpair2  4001  elop  4022  ordtri2or2exmidlem  4305  onsucelsucexmidlem  4308  en2lp  4333  reg3exmidlemwe  4357  xpsspw  4508  acexmidlem2  5588  2oconcl  6135  exmidpw  6551  renfdisj  7449  fzpr  9384  maxabslemval  10468  isprm2  10879  bj-zfpair2  11144
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