Theorem eq0

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by GG, 6-Sep-2024)

	Ref	Expression
Assertion	eq0	|- ( A = (/) <-> A. x -. x e. A )

Proof

Step	Hyp	Ref	Expression
1		dfnul4	|- (/) = { y | F. }
2	1	eqeq2i	|- ( A = (/) <-> A = { y | F. } )
3		biidd	|- ( y = x -> ( F. <-> F. ) )
4	3	eqabbw	|- ( A = { y | F. } <-> A. x ( x e. A <-> F. ) )
5		nbfal	|- ( -. x e. A <-> ( x e. A <-> F. ) )
6	5	albii	|- ( A. x -. x e. A <-> A. x ( x e. A <-> F. ) )
7	4 6	bitr4i	|- ( A = { y | F. } <-> A. x -. x e. A )
8	2 7	bitri	|- ( A = (/) <-> A. x -. x e. A )