Theorem sspwimpVD 37316

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 36939) 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 37315 is sspwimpVD 37316 without virtual deductions and was derived from sspwimpVD 37316. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

1::	|- (. A C_ B ->. A C_ B ).
2::	|- (. .............. x e. ~P A ->. x e. ~P A ).
3:2:	|- (. .............. x e. ~P A ->. x C_ A ).
4:3,1:	|- (. (. A C_ B ,. x e. ~P A ). ->. x C_ B ).
5::	|- x e. _V
6:4,5:	|- (. (. A C_ B ,. x e. ~P A ). ->. x e. ~P B ).
7:6:	|- (. A C_ B ->. ( x e. ~P A -> x e. ~P B ) ).
8:7:	|- (. A C_ B ->. A. x ( x e. ~P A -> x e. ~P B ) ).
9:8:	|- (. A C_ B ->. ~P A C_ ~P B ).
qed:9:	|- ( A C_ B -> ~P A C_ ~P B )

Assertion

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

Proof of Theorem sspwimpVD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3048	. . . . . . 7 |- x e. _V
2	1	vd01 36976	. . . . . 6 |- (. T. ->. x e. _V ).
3		idn1 36944	. . . . . . 7 |- (. A C_ B ->. A C_ B ).
4		idn1 36944	. . . . . . . 8 |- (. x e. ~P A ->. x e. ~P A ).
5		elpwi 3960	. . . . . . . 8 |- ( x e. ~P A -> x C_ A )
6	4, 5	el1 37007	. . . . . . 7 |- (. x e. ~P A ->. x C_ A ).
7		sstr 3440	. . . . . . . 8 |- ( ( x C_ A /\ A C_ B ) -> x C_ B )
8	7	ancoms 455	. . . . . . 7 |- ( ( A C_ B /\ x C_ A ) -> x C_ B )
9	3, 6, 8	el12 37113	. . . . . 6 |- (. (. A C_ B ,. x e. ~P A ). ->. x C_ B ).
10	2, 9	elpwgdedVD 37314	. . . . . 6 |- (. (. T. ,. (. A C_ B ,. x e. ~P A ). ). ->. x e. ~P B ).
11	2, 9, 10	un0.1 37166	. . . . 5 |- (. (. A C_ B ,. x e. ~P A ). ->. x e. ~P B ).
12	11	int2 36985	. . . 4 |- (. A C_ B ->. ( x e. ~P A -> x e. ~P B ) ).
13	12	gen11 36995	. . 3 |- (. A C_ B ->. A. x ( x e. ~P A -> x e. ~P B ) ).
14		dfss2 3421	. . . 4 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
15	14	biimpri 210	. . 3 |- ( A. x ( x e. ~P A -> x e. ~P B ) -> ~P A C_ ~P B )
16	13, 15	el1 37007	. 2 |- (. A C_ B ->. ~P A C_ ~P B ).
17	16	in1 36941	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   A.wal 1442   T. wtru 1445   e. wcel 1887   _Vcvv 3045   C_ wss 3404   ~Pcpw 3951   (.wvhc2 36950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-ss 3418  df-pw 3953  df-vd1 36940  df-vhc2 36951

This theorem is referenced by: (None)