Theorem sspwimpVD

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 ) using conjunction-form virtual hypothesis collections. It was completed manually, but has the potential to be completed automatically by a tools program which would invoke Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimp is sspwimpVD without virtual deductions and was derived from sspwimpVD . (Contributed by Alan Sare, 23-Apr-2015) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		vex	|- x e. _V
2	1	vd01	|- (. T. ->. x e. _V ).
3		idn1	|- (. A C_ B ->. A C_ B ).
4		idn1	|- (. x e. ~P A ->. x e. ~P A ).
5		elpwi	|- ( x e. ~P A -> x C_ A )
6	4 5	el1	|- (. 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	el12	|- (. (. A C_ B ,. x e. ~P A ). ->. x C_ B ).
10	2 9	elpwgdedVD	|- (. (. T. ,. (. A C_ B ,. x e. ~P A ). ). ->. x e. ~P B ).
11	2 9 10	un0.1	|- (. (. A C_ B ,. x e. ~P A ). ->. x e. ~P B ).
12	11	int2	|- (. A C_ B ->. ( x e. ~P A -> x e. ~P B ) ).
13	12	gen11	|- (. A C_ B ->. A. x ( x e. ~P A -> x e. ~P B ) ).
14		df-ss	|- ( ~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	el1	|- (. A C_ B ->. ~P A C_ ~P B ).
17	16	in1	|- ( A C_ B -> ~P A C_ ~P B )