Theorem unexg

Description: The union of two sets is a set. Corollary 5.8 of TakeutiZaring p. 16. (Contributed by NM, 18-Sep-2006) Prove unexg first and then unex and unexb from it. (Revised by BJ, 21-Jul-2025)

	Ref	Expression
Assertion	unexg	|- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V )

Proof

Step	Hyp	Ref	Expression
1		uniprg	|- ( ( A e. V /\ B e. W ) -> U. { A , B } = ( A u. B ) )
2		prex	|- { A , B } e. _V
3	2	a1i	|- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
4	3	uniexd	|- ( ( A e. V /\ B e. W ) -> U. { A , B } e. _V )
5	1 4	eqeltrrd	|- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V )