Theorem unexd 7598

Description: The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)

Hypotheses

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

Assertion

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

Proof of Theorem unexd

Step	Hyp	Ref	Expression
1		unexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		unexd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		unexg 7593	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2110  Vcvv 3431   ∪ cun 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-sn 4568  df-pr 4570  df-uni 4846

This theorem is referenced by:  sexp2  33789  sexp3  33795  ssltun1  33998  ssltun2  33999  ofun  40208