Theorem sspwimpcfVD

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 ) using conjunction-form virtual hypothesis collections. It was completed automatically by a tools program which would invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sspwimpcf is sspwimpcfVD without virtual deductions and was derived from sspwimpcfVD . The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		vex	|- x e. _V
2		idn1	|- (. A C_ B ->. A C_ B ).
3		idn1	|- (. x e. ~P A ->. x e. ~P A ).
4		elpwi	|- ( x e. ~P A -> x C_ A )
5	3 4	el1	|- (. 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	el12	|- (. (. 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	el021old	|- (. (. 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		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	el1	|- (. A C_ B ->. ~P A C_ ~P B ).
17	16	in1	|- ( A C_ B -> ~P A C_ ~P B )