Theorem sspwimp 37315

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. sspwimp 37315, using conventional notation, was translated from virtual deduction form, sspwimpVD 37316, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sspwimp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3048	. . . . . . 7 |- x e. _V
2	1	a1i 11	. . . . . 6 |- ( T. -> x e. _V )
3		id 22	. . . . . . 7 |- ( A C_ B -> A C_ B )
4		id 22	. . . . . . . 8 |- ( x e. ~P A -> x e. ~P A )
5		elpwi 3960	. . . . . . . 8 |- ( x e. ~P A -> x C_ A )
6	4, 5	syl 17	. . . . . . 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	syl2an 480	. . . . . 6 |- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
10	2, 9	elpwgded 36931	. . . . . 6 |- ( ( T. /\ ( A C_ B /\ x e. ~P A ) ) -> x e. ~P B )
11	2, 9, 10	uun0.1 37165	. . . . 5 |- ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
12	11	ex 436	. . . 4 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
13	12	alrimiv 1773	. . 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	syl 17	. 2 |- ( A C_ B -> ~P A C_ ~P B )
17	16	iin1 36942	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   /\ wa 371   A.wal 1442   T. wtru 1445   e. wcel 1887   _Vcvv 3045   C_ wss 3404   ~Pcpw 3951

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

This theorem is referenced by: (None)