Theorem unexg

Description: A union of two sets is a set. Corollary 5.8 of TakeutiZaring p. 16. (Contributed by NM, 18-Sep-2006)

	Ref	Expression
Assertion	unexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
2		elex	⊢ ( 𝐵 ∈ 𝑊 → 𝐵 ∈ V )
3		unexb	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ↔ ( 𝐴 ∪ 𝐵 ) ∈ V )
4	3	biimpi	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ∪ 𝐵 ) ∈ V )
5	1 2 4	syl2an	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∪ 𝐵 ) ∈ V )