Theorem elpr2OLD 4587

Description: Obsolete version of elpr2 4586 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 = 𝐵 ∨ 𝐴 = 𝐶))

Proof of Theorem elpr2OLD

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2		elpr2.1	. . . 4 ⊢ 𝐵 ∈ V
3		eleq1 2826	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
4	2, 3	mpbiri 257	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ V)
5		elpr2.2	. . . 4 ⊢ 𝐶 ∈ V
6		eleq1 2826	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
7	5, 6	mpbiri 257	. . 3 ⊢ (𝐴 = 𝐶 → 𝐴 ∈ V)
8	4, 7	jaoi 854	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → 𝐴 ∈ V)
9		elprg 4582	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
10	1, 8, 9	pm5.21nii 380	1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))

Syntax hints:   ↔ wb 205   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-sn 4562  df-pr 4564

This theorem is referenced by: (None)