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Metamath Proof Explorer

Theorem dfnul4

Description: Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019) Avoid ax-8 , df-clel . (Revised by GG, 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. }