Theorem ab0

Description: The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne ). (Contributed by BJ, 19-Mar-2021)

	Ref	Expression
Assertion	ab0	|- ( { x | ph } = (/) <-> A. x -. ph )

Proof

Step	Hyp	Ref	Expression
1		nfab1	|- F/_ x { x | ph }
2	1	eq0f	|- ( { x | ph } = (/) <-> A. x -. x e. { x | ph } )
3		abid	|- ( x e. { x | ph } <-> ph )
4	3	notbii	|- ( -. x e. { x | ph } <-> -. ph )
5	4	albii	|- ( A. x -. x e. { x | ph } <-> A. x -. ph )
6	2 5	bitri	|- ( { x | ph } = (/) <-> A. x -. ph )