Theorem elpwunicl

Description: Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020) (Proof shortened by BJ, 6-Apr-2024)

	Ref	Expression
Hypothesis	elpwunicl.1	$$⊢\phi \to A\in 𝒫𝒫B$$
Assertion	elpwunicl	$$⊢\phi \to \bigcup A\in 𝒫B$$

Proof

Step	Hyp	Ref	Expression
1		elpwunicl.1	$$⊢\phi \to A\in 𝒫𝒫B$$
2		elpwpwel	$$⊢A\in 𝒫𝒫B\leftrightarrow \bigcup A\in 𝒫B$$
3	1 2	sylib	$$⊢\phi \to \bigcup A\in 𝒫B$$