Theorem elpr2OLD 4654

Description: Obsolete version of elpr2 4653 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 3492	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2		elpr2.1	. . . 4 ⊢ 𝐵 ∈ V
3		eleq1 2820	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ V)
5		elpr2.2	. . . 4 ⊢ 𝐶 ∈ V
6		eleq1 2820	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
7	5, 6	mpbiri 258	. . 3 ⊢ (𝐴 = 𝐶 → 𝐴 ∈ V)
8	4, 7	jaoi 854	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → 𝐴 ∈ V)
9		elprg 4649	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
10	1, 8, 9	pm5.21nii 378	1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))

Syntax hints:   ↔ wb 205   ∨ wo 844   = wceq 1540   ∈ wcel 2105  Vcvv 3473  {cpr 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-sn 4629  df-pr 4631

This theorem is referenced by: (None)