Theorem bj-0nelmpt

Description: The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020)

	Ref	Expression
Assertion	bj-0nelmpt	|- -. (/) e. ( x e. A |-> B )

Proof

Step	Hyp	Ref	Expression
1		0nelopab	|- -. (/) e. { <. x , y >. | ( x e. A /\ y = B ) }
2		df-mpt	|- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
3	2	eqcomi	|- { <. x , y >. | ( x e. A /\ y = B ) } = ( x e. A |-> B )
4	3	eleq2i	|- ( (/) e. { <. x , y >. | ( x e. A /\ y = B ) } <-> (/) e. ( x e. A |-> B ) )
5	1 4	mtbi	|- -. (/) e. ( x e. A |-> B )