Theorem nbgr0vtx

Description: In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020) (Proof shortened by AV, 10-May-2025)

	Ref	Expression
Assertion	nbgr0vtx	|- ( ( Vtx ` G ) = (/) -> ( G NeighbVtx K ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		nel02	|- ( ( Vtx ` G ) = (/) -> -. K e. ( Vtx ` G ) )
2		df-nel	|- ( K e/ ( Vtx ` G ) <-> -. K e. ( Vtx ` G ) )
3	1 2	sylibr	|- ( ( Vtx ` G ) = (/) -> K e/ ( Vtx ` G ) )
4		eqid	|- ( Vtx ` G ) = ( Vtx ` G )
5	4	nbgrnvtx0	|- ( K e/ ( Vtx ` G ) -> ( G NeighbVtx K ) = (/) )
6	3 5	syl	|- ( ( Vtx ` G ) = (/) -> ( G NeighbVtx K ) = (/) )