Theorem sspwimpVD 39155

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 38785) 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 39154 is sspwimpVD 39155 without virtual deductions and was derived from sspwimpVD 39155. (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 3203	. . . . . . 7 |- x e. _V
2	1	vd01 38822	. . . . . 6 |- (. T. ->. x e. _V ).
3		idn1 38790	. . . . . . 7 |- (. A C_ B ->. A C_ B ).
4		idn1 38790	. . . . . . . 8 |- (. x e. ~P A ->. x e. ~P A ).
5		elpwi 4168	. . . . . . . 8 |- ( x e. ~P A -> x C_ A )
6	4, 5	el1 38853	. . . . . . 7 |- (. x e. ~P A ->. x C_ A ).
7		sstr 3611	. . . . . . . 8 |- ( ( x C_ A /\ A C_ B ) -> x C_ B )
8	7	ancoms 469	. . . . . . 7 |- ( ( A C_ B /\ x C_ A ) -> x C_ B )
9	3, 6, 8	el12 38953	. . . . . 6 |- (. (. A C_ B ,. x e. ~P A ). ->. x C_ B ).
10	2, 9	elpwgdedVD 39153	. . . . . 6 |- (. (. T. ,. (. A C_ B ,. x e. ~P A ). ). ->. x e. ~P B ).
11	2, 9, 10	un0.1 39006	. . . . 5 |- (. (. A C_ B ,. x e. ~P A ). ->. x e. ~P B ).
12	11	int2 38831	. . . 4 |- (. A C_ B ->. ( x e. ~P A -> x e. ~P B ) ).
13	12	gen11 38841	. . 3 |- (. A C_ B ->. A. x ( x e. ~P A -> x e. ~P B ) ).
14		dfss2 3591	. . . 4 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
15	14	biimpri 218	. . 3 |- ( A. x ( x e. ~P A -> x e. ~P B ) -> ~P A C_ ~P B )
16	13, 15	el1 38853	. 2 |- (. A C_ B ->. ~P A C_ ~P B ).
17	16	in1 38787	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   A.wal 1481   T. wtru 1484   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   (.wvhc2 38796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-vd1 38786  df-vhc2 38797

This theorem is referenced by: (None)