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)

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

Proof

Step	Hyp	Ref	Expression
1		df-nul	|- (/) = ( _V \ _V )
2		df-dif	|- ( _V \ _V ) = { x | ( x e. _V /\ -. x e. _V ) }
3		pm3.24	|- -. ( x e. _V /\ -. x e. _V )
4		equid	|- x = x
5	4	notnoti	|- -. -. x = x
6	3 5	2false	|- ( ( x e. _V /\ -. x e. _V ) <-> -. x = x )
7	6	abbii	|- { x | ( x e. _V /\ -. x e. _V ) } = { x | -. x = x }
8	1 2 7	3eqtri	|- (/) = { x | -. x = x }