Theorem bnj1149 32277

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 7463	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 588	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2112  Vcvv 3407   ∪ cun 3852

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3077  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-sn 4516  df-pr 4518  df-uni 4792

This theorem is referenced by:  bnj1136  32482  bnj1413  32520  bnj1452  32537  bnj1489  32541