Theorem uniexr

Description: Converse of the Axiom of Union. Note that it does not require ax-un . (Contributed by NM, 11-Nov-2003)

	Ref	Expression
Assertion	uniexr	$$⊢\bigcup A\in V\to A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		pwuni	$$⊢A\subseteq 𝒫\bigcup A$$
2		pwexg	$$⊢\bigcup A\in V\to 𝒫\bigcup A\in \mathrm{V}$$
3		ssexg	$$⊢\left(A\subseteq 𝒫\bigcup A\wedge 𝒫\bigcup A\in \mathrm{V}\right)\to A\in \mathrm{V}$$
4	1 2 3	sylancr	$$⊢\bigcup A\in V\to A\in \mathrm{V}$$