Theorem dfnul2

Description: Alternate definition of the empty set. Definition 5.14 of TakeutiZaring p. 20. (Contributed by NM, 26-Dec-1996) Remove dependency on ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 3-May-2023) (Proof shortened by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	dfnul2	|- (/) = { x | -. x = x }

Proof

Step	Hyp	Ref	Expression
1		dfnul4	|- (/) = { x | F. }
2		equid	|- x = x
3	2	notnoti	|- -. -. x = x
4	3	bifal	|- ( -. x = x <-> F. )
5	4	abbii	|- { x | -. x = x } = { x | F. }
6	1 5	eqtr4i	|- (/) = { x | -. x = x }