Theorem sspwimp

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. For the biconditional, see sspwb . The proof sspwimp , using conventional notation, was translated from virtual deduction form, sspwimpVD , using a translation program. (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		vex	|- x e. _V
2	1	a1i	|- ( T. -> x e. _V )
3		id	|- ( A C_ B -> A C_ B )
4		id	|- ( x e. ~P A -> x e. ~P A )
5		elpwi	|- ( x e. ~P A -> x C_ A )
6	4 5	syl	|- ( x e. ~P A -> x C_ A )
7		sstr	|- ( ( x C_ A /\ A C_ B ) -> x C_ B )
8	7	ancoms	|- ( ( A C_ B /\ x C_ A ) -> x C_ B )
9	3 6 8	syl2an	|- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
10	2 9	elpwgded	|- ( ( T. /\ ( A C_ B /\ x e. ~P A ) ) -> x e. ~P B )
11	2 9 10	uun0.1	|- ( ( 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 )