Theorem bnj1149 35489

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 7781	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4871

This theorem is used by:  bnj1136  35694  bnj1413  35732  bnj1452  35749  bnj1489  35753