Theorem bnj1149 31380

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1149.1	⊢ (𝜑 → 𝐴 ∈ V)
bnj1149.2	⊢ (𝜑 → 𝐵 ∈ V)

Assertion

Ref	Expression
bnj1149	⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Proof of Theorem bnj1149

Step	Hyp	Ref	Expression
1		bnj1149.1	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
2		bnj1149.2	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
3		unexg 7193	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 580	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2157  Vcvv 3385   ∪ cun 3767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-sn 4369  df-pr 4371  df-uni 4629

This theorem is referenced by:  bnj1136  31582  bnj1413  31620  bnj1452  31637  bnj1489  31641