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Theorem elpr2 4141
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, 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 elex 3179 . 2 (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2 elpr2.1 . . . 4 𝐵 ∈ V
3 eleq1 2670 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
42, 3mpbiri 246 . . 3 (𝐴 = 𝐵𝐴 ∈ V)
5 elpr2.2 . . . 4 𝐶 ∈ V
6 eleq1 2670 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
75, 6mpbiri 246 . . 3 (𝐴 = 𝐶𝐴 ∈ V)
84, 7jaoi 392 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
9 elprg 4138 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
101, 8, 9pm5.21nii 366 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wo 381   = wceq 1474  wcel 1975  Vcvv 3167  {cpr 4121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-v 3169  df-un 3539  df-sn 4120  df-pr 4122
This theorem is referenced by:  elopg  4850  elxr  11780  nofv  30855
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