Theorem unexgOLD 7682

Description: Obsolete version of unexg 7676 as of 21-Jul-2025. (Contributed by NM, 18-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
unexgOLD	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem unexgOLD

Step	Hyp	Ref	Expression
1		elex 3457	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3457	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		unexb 7680	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∪ 𝐵) ∈ V)
4	3	biimpi 216	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
5	1, 2, 4	syl2an 596	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-sn 4574  df-pr 4576  df-uni 4857

This theorem is referenced by: (None)