Theorem canth2g

Description: Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of Suppes p. 97. (Contributed by NM, 7-Nov-2003)

	Ref	Expression
Assertion	canth2g	|- ( A e. V -> A ~< ~P A )

Proof

Step	Hyp	Ref	Expression
1		pweq	|- ( x = A -> ~P x = ~P A )
2		breq12	|- ( ( x = A /\ ~P x = ~P A ) -> ( x ~< ~P x <-> A ~< ~P A ) )
3	1 2	mpdan	|- ( x = A -> ( x ~< ~P x <-> A ~< ~P A ) )
4		vex	|- x e. _V
5	4	canth2	|- x ~< ~P x
6	3 5	vtoclg	|- ( A e. V -> A ~< ~P A )