Theorem elpr2g 4582

Description: A member of a pair of sets is one or the other of them, and conversely. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) Generalize from sethood hypothesis to sethood antecedent. (Revised by BJ, 25-May-2024.)

Assertion

Ref	Expression
elpr2g	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))

Proof of Theorem elpr2g

Step	Hyp	Ref	Expression
1		elex 3440	. . 3 ⊢ (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2	1	a1i 11	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V))
3		elex 3440	. . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
4		eleq1a 2834	. . . 4 ⊢ (𝐵 ∈ V → (𝐴 = 𝐵 → 𝐴 ∈ V))
5	3, 4	syl 17	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ∈ V))
6		elex 3440	. . . 4 ⊢ (𝐶 ∈ 𝑊 → 𝐶 ∈ V)
7		eleq1a 2834	. . . 4 ⊢ (𝐶 ∈ V → (𝐴 = 𝐶 → 𝐴 ∈ V))
8	6, 7	syl 17	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐴 = 𝐶 → 𝐴 ∈ V))
9	5, 8	jaao 951	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → 𝐴 ∈ V))
10		elprg 4579	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
11	10	a1i 11	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))))
12	2, 9, 11	pm5.21ndd 380	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-sn 4559  df-pr 4561

This theorem is referenced by:  elpr2  4583