Theorem unex

Description: The union of two sets is a set. Corollary 5.8 of TakeutiZaring p. 16. (Contributed by NM, 1-Jul-1994)

	Ref	Expression
Hypotheses	unex.1	⊢ 𝐴 ∈ V
	unex.2	⊢ 𝐵 ∈ V
Assertion	unex	⊢ ( 𝐴 ∪ 𝐵 ) ∈ V

Proof

Step	Hyp	Ref	Expression
1		unex.1	⊢ 𝐴 ∈ V
2		unex.2	⊢ 𝐵 ∈ V
3	1 2	unipr	⊢ ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 )
4		prex	⊢ { 𝐴 , 𝐵 } ∈ V
5	4	uniex	⊢ ∪ { 𝐴 , 𝐵 } ∈ V
6	3 5	eqeltrri	⊢ ( 𝐴 ∪ 𝐵 ) ∈ V