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Theorem elpr 3385
 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 3384 . 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 628   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {cpr 3368 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374 This theorem is referenced by:  prmg  3480  difprsnss  3493  preqr1  3530  preq12b  3532  prel12  3533  pwprss  3567  pwtpss  3568  unipr  3585  intpr  3638  zfpair2  3936  elop  3959  onsucelsucexmidlem  4214  en2lp  4232  xpsspw  4393  acexmidlem2  5452  2oconcl  5961  renfdisj  6876  fzpr  8709  bj-zfpair2  9365
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