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Metamath Proof Explorer

Theorem sspwimpcf

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

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

Proof

Step Hyp Ref Expression
1 vex 
 |-  x e. _V
2 id 
 |-  ( A C_ B -> A C_ B )
3 id 
 |-  ( x e. ~P A -> x e. ~P A )
4 elpwi 
 |-  ( x e. ~P A -> x C_ A )
5 3 4 syl 
 |-  ( x e. ~P A -> x C_ A )
6 sstr2 
 |-  ( x C_ A -> ( A C_ B -> x C_ B ) )
7 6 impcom 
 |-  ( ( A C_ B /\ x C_ A ) -> x C_ B )
8 2 5 7 syl2an 
 |-  ( ( A C_ B /\ x e. ~P A ) -> x C_ B )
9 elpwg 
 |-  ( x e. _V -> ( x e. ~P B <-> x C_ B ) )
10 9 biimpar 
 |-  ( ( x e. _V /\ x C_ B ) -> x e. ~P B )
11 1 8 10 eel021old 
 |-  ( ( A C_ B /\ x e. ~P A ) -> x e. ~P B )
12 11 ex 
 |-  ( A C_ B -> ( x e. ~P A -> x e. ~P B ) )
13 12 alrimiv 
 |-  ( A C_ B -> A. x ( x e. ~P A -> x e. ~P B ) )
14 df-ss 
 |-  ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
15 14 biimpri 
 |-  ( A. x ( x e. ~P A -> x e. ~P B ) -> ~P A C_ ~P B )
16 13 15 syl 
 |-  ( A C_ B -> ~P A C_ ~P B )
17 16 iin1 
 |-  ( A C_ B -> ~P A C_ ~P B )