Theorem sspwimpALT 39161

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. sspwimpALT 39161 is the completed proof in conventional notation of the Virtual Deduction proof http://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 38785 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 9 used elpwgded 38780). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 5 used elpwi 4168). (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sspwimpALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . . 8 |- x e. _V
2	1	a1i 11	. . . . . . 7 |- ( T. -> x e. _V )
3		id 22	. . . . . . . . 9 |- ( x e. ~P A -> x e. ~P A )
4		elpwi 4168	. . . . . . . . 9 |- ( x e. ~P A -> x C_ A )
5	3, 4	syl 17	. . . . . . . 8 |- ( x e. ~P A -> x C_ A )
6		id 22	. . . . . . . 8 |- ( A C_ B -> A C_ B )
7	5, 6	sylan9ssr 3617	. . . . . . 7 |- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
8	2, 7	elpwgded 38780	. . . . . 6 |- ( ( T. /\ ( A C_ B /\ x e. ~P A ) ) -> x e. ~P B )
9	8	uunT1 39007	. . . . 5 |- ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
10	9	ex 450	. . . 4 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
11	10	alrimiv 1855	. . 3 |- ( A C_ B -> A. x ( x e. ~P A -> x e. ~P B ) )
12		dfss2 3591	. . . 4 |- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
13	12	biimpri 218	. . 3 |- ( A. x ( x e. ~P A -> x e. ~P B ) -> ~P A C_ ~P B )
14	11, 13	syl 17	. 2 |- ( A C_ B -> ~P A C_ ~P B )
15	14	idiALT 38683	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   T. wtru 1484   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)