Theorem elpwd

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	elpwd.1	|- ( ph -> A e. V )
	elpwd.2	|- ( ph -> A C_ B )
Assertion	elpwd	|- ( ph -> A e. ~P B )

Proof

Step	Hyp	Ref	Expression
1		elpwd.1	|- ( ph -> A e. V )
2		elpwd.2	|- ( ph -> A C_ B )
3		elpwg	|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )
4	1 3	syl	|- ( ph -> ( A e. ~P B <-> A C_ B ) )
5	2 4	mpbird	|- ( ph -> A e. ~P B )