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	$$⊢\left(A\in V\wedge B\in W\right)\to A\cup B\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in V\to A\in \mathrm{V}$$
2		elex	$$⊢B\in W\to B\in \mathrm{V}$$
3		unexb	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\leftrightarrow A\cup B\in \mathrm{V}$$
4	3	biimpi	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to A\cup B\in \mathrm{V}$$
5	1 2 4	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to A\cup B\in \mathrm{V}$$