Theorem sspwimpALT

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. sspwimpALT is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html . It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 9 used elpwgded ). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 5 used elpwi ). (Contributed by Alan Sare, 3-Dec-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sspwimpALT	|- ( 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	|- ( 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		id	|- ( A C_ B -> A C_ B )
7	5 6	sylan9ssr	|- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
8	2 7	elpwgded	|- ( ( T. /\ ( A C_ B /\ x e. ~P A ) ) -> x e. ~P B )
9	8	uunT1	|- ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
10	9	ex	|- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
11	10	alrimiv	|- ( A C_ B -> A. x ( x e. ~P A -> x e. ~P B ) )
12		dfss2	|- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
13	12	biimpri	|- ( A. x ( x e. ~P A -> x e. ~P B ) -> ~P A C_ ~P B )
14	11 13	syl	|- ( A C_ B -> ~P A C_ ~P B )
15	14	idiALT	|- ( A C_ B -> ~P A C_ ~P B )