Theorem psspwb

Description: Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022)

	Ref	Expression
Assertion	psspwb	|- ( A C. B <-> ~P A C. ~P B )

Proof

Step	Hyp	Ref	Expression
1		sspwb	|- ( A C_ B <-> ~P A C_ ~P B )
2		pweqb	|- ( A = B <-> ~P A = ~P B )
3	2	necon3bii	|- ( A =/= B <-> ~P A =/= ~P B )
4	1 3	anbi12i	|- ( ( A C_ B /\ A =/= B ) <-> ( ~P A C_ ~P B /\ ~P A =/= ~P B ) )
5		df-pss	|- ( A C. B <-> ( A C_ B /\ A =/= B ) )
6		df-pss	|- ( ~P A C. ~P B <-> ( ~P A C_ ~P B /\ ~P A =/= ~P B ) )
7	4 5 6	3bitr4i	|- ( A C. B <-> ~P A C. ~P B )