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Metamath Proof Explorer

Theorem sspwb

Description: The powerclass construction preserves and reflects inclusion. Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of TakeutiZaring p. 18. (Contributed by NM, 13-Oct-1996)

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

Proof

Step Hyp Ref Expression
1 sspw 
 |-  ( A C_ B -> ~P A C_ ~P B )
2 ssel 
 |-  ( ~P A C_ ~P B -> ( { x } e. ~P A -> { x } e. ~P B ) )
3 vsnex 
 |-  { x } e. _V
4 3 elpw 
 |-  ( { x } e. ~P A <-> { x } C_ A )
5 vex 
 |-  x e. _V
6 5 snss 
 |-  ( x e. A <-> { x } C_ A )
7 4 6 bitr4i 
 |-  ( { x } e. ~P A <-> x e. A )
8 3 elpw 
 |-  ( { x } e. ~P B <-> { x } C_ B )
9 5 snss 
 |-  ( x e. B <-> { x } C_ B )
10 8 9 bitr4i 
 |-  ( { x } e. ~P B <-> x e. B )
11 2 7 10 3imtr3g 
 |-  ( ~P A C_ ~P B -> ( x e. A -> x e. B ) )
12 11 ssrdv 
 |-  ( ~P A C_ ~P B -> A C_ B )
13 1 12 impbii 
 |-  ( A C_ B <-> ~P A C_ ~P B )