Theorem nbgr0edg

Description: In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 26-Oct-2020) (Proof shortened by AV, 15-Nov-2020)

	Ref	Expression
Assertion	nbgr0edg	|- ( ( Edg ` G ) = (/) -> ( G NeighbVtx K ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		rzal	|- ( ( Edg ` G ) = (/) -> A. e e. ( Edg ` G ) -. { K , n } C_ e )
2		ralnex	|- ( A. e e. ( Edg ` G ) -. { K , n } C_ e <-> -. E. e e. ( Edg ` G ) { K , n } C_ e )
3	1 2	sylib	|- ( ( Edg ` G ) = (/) -> -. E. e e. ( Edg ` G ) { K , n } C_ e )
4	3	ralrimivw	|- ( ( Edg ` G ) = (/) -> A. n e. ( ( Vtx ` G ) \ { K } ) -. E. e e. ( Edg ` G ) { K , n } C_ e )
5	4	nbgr0vtxlem	|- ( ( Edg ` G ) = (/) -> ( G NeighbVtx K ) = (/) )