Theorem sspwimpALT2 39164

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. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in http://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sspwimpALT2	|- ( A C_ B -> ~P A C_ ~P B )

Proof of Theorem sspwimpALT2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 |- x e. _V
2		elpwi 4168	. . . . 5 |- ( x e. ~P A -> x C_ A )
3		id 22	. . . . 5 |- ( A C_ B -> A C_ B )
4	2, 3	sylan9ssr 3617	. . . 4 |- ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
5		elpwg 4166	. . . . 5 |- ( x e. _V -> ( x e. ~P B <-> x C_ B ) )
6	5	biimpar 502	. . . 4 |- ( ( x e. _V /\ x C_ B ) -> x e. ~P B )
7	1, 4, 6	sylancr 695	. . 3 |- ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
8	7	ex 450	. 2 |- ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
9	8	ssrdv 3609	1 |- ( A C_ B -> ~P A C_ ~P B )

Syntax hints:   -> wi 4   /\ wa 384   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)