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Theorem elpr2OLD 4554
Description: Obsolete version of elpr2 4553 as of 25-May-2024. (Contributed by NM, 14-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpr2.1 𝐵 ∈ V
elpr2.2 𝐶 ∈ V
Assertion
Ref Expression
elpr2OLD (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpr2OLD
StepHypRef Expression
1 elex 3462 . 2 (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2 elpr2.1 . . . 4 𝐵 ∈ V
3 eleq1 2880 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
42, 3mpbiri 261 . . 3 (𝐴 = 𝐵𝐴 ∈ V)
5 elpr2.2 . . . 4 𝐶 ∈ V
6 eleq1 2880 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
75, 6mpbiri 261 . . 3 (𝐴 = 𝐶𝐴 ∈ V)
84, 7jaoi 854 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
9 elprg 4549 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
101, 8, 9pm5.21nii 383 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wo 844   = wceq 1538  wcel 2112  Vcvv 3444  {cpr 4530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-sn 4529  df-pr 4531
This theorem is referenced by: (None)
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