Theorem sspwimpcf 39156

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. sspwimpcf 39156, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 39157, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sspwimpcf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- x e. _V
2		id 22	. . . . . . 7 |- ( A C_ B -> A C_ B )
3		id 22	. . . . . . . 8 |- ( x e. ~P A -> x e. ~P A )
4		elpwi 4168	. . . . . . . 8 |- ( x e. ~P A -> x C_ A )
5	3, 4	syl 17	. . . . . . 7 |- ( x e. ~P A -> x C_ A )
6		sstr2 3610	. . . . . . . 8 |- ( x C_ A -> ( A C_ B -> x C_ B ) )
7	6	impcom 446	. . . . . . 7 |- ( ( A C_ B /\ x C_ A ) -> x C_ B )
8	2, 5, 7	syl2an 494	. . . . . 6 |- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
9		elpwg 4166	. . . . . . 7 |- ( x e. _V -> ( x e. ~P B <-> x C_ B ) )
10	9	biimpar 502	. . . . . 6 |- ( ( x e. _V /\ x C_ B ) -> x e. ~P B )
11	1, 8, 10	eel021old 38925	. . . . 5 |- ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
12	11	ex 450	. . . 4 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
13	12	alrimiv 1855	. . 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	syl 17	. 2 |- ( A C_ B -> ~P A C_ ~P B )
17	16	iin1 38788	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158

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

This theorem is referenced by: (None)