Theorem elpwgdedVD

Description: Membership in a power class. Theorem 86 of Suppes p. 47. Derived from elpwg . In form of VD deduction with ph and ps as variable virtual hypothesis collections based on Mario Carneiro's metavariable concept. elpwgded is elpwgdedVD using conventional notation. (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	elpwgdedVD.1	|- (. ph ->. A e. _V ).
	elpwgdedVD.2	|- (. ps ->. A C_ B ).
Assertion	elpwgdedVD	|- (. (. ph ,. ps ). ->. A e. ~P B ).

Proof

Step	Hyp	Ref	Expression
1		elpwgdedVD.1	|- (. ph ->. A e. _V ).
2		elpwgdedVD.2	|- (. ps ->. A C_ B ).
3		elpwg	|- ( A e. _V -> ( A e. ~P B <-> A C_ B ) )
4	3	biimpar	|- ( ( A e. _V /\ A C_ B ) -> A e. ~P B )
5	1 2 4	el12	|- (. (. ph ,. ps ). ->. A e. ~P B ).