Theorem bnj1149 32772

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 7599	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885

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  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-uni 4840

This theorem is referenced by:  bnj1136  32977  bnj1413  33015  bnj1452  33032  bnj1489  33036