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		snex	|- { 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 )