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	|- ( ph -> A e. ~P ~P B )
Assertion	elpwunicl	|- ( ph -> U. A e. ~P B )

Proof

Step	Hyp	Ref	Expression
1		elpwunicl.1	|- ( ph -> A e. ~P ~P B )
2		elpwpwel	|- ( A e. ~P ~P B <-> U. A e. ~P B )
3	1 2	sylib	|- ( ph -> U. A e. ~P B )