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) Avoid df-clel , ax-8 . (Revised by Gino Giotto, 30-Aug-2024) (Proof shortened by SN, 8-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		abbi	|- ( A. x ( ph <-> F. ) <-> { x | ph } = { x | F. } )
2		nbfal	|- ( -. ph <-> ( ph <-> F. ) )
3	2	albii	|- ( A. x -. ph <-> A. x ( ph <-> F. ) )
4		dfnul4	|- (/) = { x | F. }
5	4	eqeq2i	|- ( { x | ph } = (/) <-> { x | ph } = { x | F. } )
6	1 3 5	3bitr4ri	|- ( { x | ph } = (/) <-> A. x -. ph )