Theorem dfnul4

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019) Avoid ax-8 , df-clel . (Revised by Gino Giotto, 3-Sep-2024) Prove directly from definition to allow shortening dfnul2 . (Revised by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	dfnul4	|- (/) = { x | F. }

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	3	bifal	|- ( ( x e. _V /\ -. x e. _V ) <-> F. )
5	4	abbii	|- { x | ( x e. _V /\ -. x e. _V ) } = { x | F. }
6	1 2 5	3eqtri	|- (/) = { x | F. }