Theorem pwnss

Description: The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015)

	Ref	Expression
Assertion	pwnss	|- ( A e. V -> -. ~P A C_ A )

Proof

Step	Hyp	Ref	Expression
1		rru	|- -. { x e. A | -. x e. x } e. A
2		ssel	|- ( ~P A C_ A -> ( { x e. A | -. x e. x } e. ~P A -> { x e. A | -. x e. x } e. A ) )
3	1 2	mtoi	|- ( ~P A C_ A -> -. { x e. A | -. x e. x } e. ~P A )
4		ssrab2	|- { x e. A | -. x e. x } C_ A
5		elpw2g	|- ( A e. V -> ( { x e. A | -. x e. x } e. ~P A <-> { x e. A | -. x e. x } C_ A ) )
6	4 5	mpbiri	|- ( A e. V -> { x e. A | -. x e. x } e. ~P A )
7	3 6	nsyl3	|- ( A e. V -> -. ~P A C_ A )