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 Gino Giotto and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by Gino Giotto, 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		dfcleq	|- ( A = { y | F. } <-> A. x ( x e. A <-> x e. { y | F. } ) )
4		df-clab	|- ( x e. { y | F. } <-> [ x / y ] F. )
5		sbv	|- ( [ x / y ] F. <-> F. )
6	4 5	bitri	|- ( x e. { y | F. } <-> F. )
7	6	bibi2i	|- ( ( x e. A <-> x e. { y | F. } ) <-> ( x e. A <-> F. ) )
8		nbfal	|- ( -. x e. A <-> ( x e. A <-> F. ) )
9	7 8	bitr4i	|- ( ( x e. A <-> x e. { y | F. } ) <-> -. x e. A )
10	9	albii	|- ( A. x ( x e. A <-> x e. { y | F. } ) <-> A. x -. x e. A )
11	3 10	bitri	|- ( A = { y | F. } <-> A. x -. x e. A )
12	2 11	bitri	|- ( A = (/) <-> A. x -. x e. A )