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Theorem elpr2OLD 4649
Description: Obsolete version of elpr2 4648 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 3487 . 2 (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2 elpr2.1 . . . 4 𝐵 ∈ V
3 eleq1 2815 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
42, 3mpbiri 258 . . 3 (𝐴 = 𝐵𝐴 ∈ V)
5 elpr2.2 . . . 4 𝐶 ∈ V
6 eleq1 2815 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
75, 6mpbiri 258 . . 3 (𝐴 = 𝐶𝐴 ∈ V)
84, 7jaoi 854 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
9 elprg 4644 . 2 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
101, 8, 9pm5.21nii 378 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844   = wceq 1533  wcel 2098  Vcvv 3468  {cpr 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948  df-sn 4624  df-pr 4626
This theorem is referenced by: (None)
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