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Theorem elpr2 4591
Description: A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) (Proof shortened by JJ, 23-Jul-2021.)
Hypotheses
Ref Expression
elpr2.1 𝐵 ∈ V
elpr2.2 𝐶 ∈ V
Assertion
Ref Expression
elpr2 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpr2
StepHypRef Expression
1 elpr2.1 . 2 𝐵 ∈ V
2 elpr2.2 . 2 𝐶 ∈ V
3 elpr2g 4590 . 2 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
41, 2, 3mp2an 688 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 843   = wceq 1541  wcel 2109  Vcvv 3430  {cpr 4568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-un 3896  df-sn 4567  df-pr 4569
This theorem is referenced by:  elopg  5383  elxr  12834  fprodex01  31118  nofv  33839
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