Theorem sspwimp 33861

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 33861, using conventional notation, was translated from virtual deduction form, sspwimpVD 33862, 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 3112	. . . . . . 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 4024	. . . . . . . 8 |- ( x e. ~P A -> x C_ A )
6	4, 5	syl 16	. . . . . . 7 |- ( x e. ~P A -> x C_ A )
7		sstr 3507	. . . . . . . 8 |- ( ( x C_ A /\ A C_ B ) -> x C_ B )
8	7	ancoms 453	. . . . . . 7 |- ( ( A C_ B /\ x C_ A ) -> x C_ B )
9	3, 6, 8	syl2an 477	. . . . . 6 |- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
10	2, 9	elpwgded 33480	. . . . . 6 |- ( ( T. /\ ( A C_ B /\ x e. ~P A ) ) -> x e. ~P B )
11	2, 9, 10	uun0.1 33718	. . . . 5 |- ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
12	11	ex 434	. . . 4 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
13	12	alrimiv 1720	. . 3 |- ( A C_ B -> A. x ( x e. ~P A -> x e. ~P B ) )
14		dfss2 3488	. . . 4 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
15	14	biimpri 206	. . 3 |- ( A. x ( x e. ~P A -> x e. ~P B ) -> ~P A C_ ~P B )
16	13, 15	syl 16	. 2 |- ( A C_ B -> ~P A C_ ~P B )
17	16	iin1 33492	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1393   T. wtru 1396   e. wcel 1819   _Vcvv 3109   C_ wss 3471   ~Pcpw 4015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017

This theorem is referenced by: (None)