Theorem sspwimpcf

Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of TakeutiZaring p. 18. sspwimpcf , using conventional notation, was translated from its virtual deduction form, sspwimpcfVD , using a translation program. (Contributed by Alan Sare, 13-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sspwimpcf	|- ( A C_ B -> ~P A C_ ~P B )

Proof

Step	Hyp	Ref	Expression
1		vex	|- x e. _V
2		id	|- ( A C_ B -> A C_ B )
3		id	|- ( x e. ~P A -> x e. ~P A )
4		elpwi	|- ( x e. ~P A -> x C_ A )
5	3 4	syl	|- ( x e. ~P A -> x C_ A )
6		sstr2	|- ( x C_ A -> ( A C_ B -> x C_ B ) )
7	6	impcom	|- ( ( A C_ B /\ x C_ A ) -> x C_ B )
8	2 5 7	syl2an	|- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
9		elpwg	|- ( x e. _V -> ( x e. ~P B <-> x C_ B ) )
10	9	biimpar	|- ( ( x e. _V /\ x C_ B ) -> x e. ~P B )
11	1 8 10	eel021old	|- ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
12	11	ex	|- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
13	12	alrimiv	|- ( A C_ B -> A. x ( x e. ~P A -> x e. ~P B ) )
14		dfss2	|- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
15	14	biimpri	|- ( A. x ( x e. ~P A -> x e. ~P B ) -> ~P A C_ ~P B )
16	13 15	syl	|- ( A C_ B -> ~P A C_ ~P B )
17	16	iin1	|- ( A C_ B -> ~P A C_ ~P B )