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Theorem elpr2 4232
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 3243 . 2 (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2 elpr2.1 . . . 4 𝐵 ∈ V
3 eleq1 2718 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
42, 3mpbiri 248 . . 3 (𝐴 = 𝐵𝐴 ∈ V)
5 elpr2.2 . . . 4 𝐶 ∈ V
6 eleq1 2718 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
75, 6mpbiri 248 . . 3 (𝐴 = 𝐶𝐴 ∈ V)
84, 7jaoi 393 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
9 elprg 4229 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
101, 8, 9pm5.21nii 367 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382   = wceq 1523  wcel 2030  Vcvv 3231  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213
This theorem is referenced by:  elopg  4964  elxr  11988  fprodex01  29699  nofv  31935
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