Metamath Proof Explorer


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 )