Theorem bnj1149 32672

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 7577	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881

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  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561  df-uni 4837

This theorem is referenced by:  bnj1136  32877  bnj1413  32915  bnj1452  32932  bnj1489  32936