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Theorem elpr 3368
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 = B A = 𝐶))

Proof of Theorem elpr
StepHypRef Expression
1 elpr.1 . 2 A V
2 elprg 3367 . 2 (A V → (A {B, 𝐶} ↔ (A = B A = 𝐶)))
31, 2ax-mp 7 1 (A {B, 𝐶} ↔ (A = B A = 𝐶))
Colors of variables: wff set class
Syntax hints:  wb 98   wo 616   = wceq 1228   wcel 1374  Vcvv 2535  {cpr 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357
This theorem is referenced by:  prmg  3463  difprsnss  3476  preqr1  3513  preq12b  3515  prel12  3516  pwprss  3550  pwtpss  3551  unipr  3568  intpr  3621  zfpair2  3919  elop  3942  onsucelsucexmidlem  4198  en2lp  4216  xpsspw  4377  acexmidlem2  5433  2oconcl  5937  renfdisj  6680  bj-zfpair2  7133
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