Theorem nbgrisvtx

Description: Every neighbor N of a vertex K is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 26-Oct-2020) (Revised by AV, 12-Feb-2022)

	Ref	Expression
Hypothesis	nbgrisvtx.v	|- V = ( Vtx ` G )
Assertion	nbgrisvtx	|- ( N e. ( G NeighbVtx K ) -> N e. V )

Proof

Step	Hyp	Ref	Expression
1		nbgrisvtx.v	|- V = ( Vtx ` G )
2		eqid	|- ( Edg ` G ) = ( Edg ` G )
3	1 2	nbgrel	|- ( N e. ( G NeighbVtx K ) <-> ( ( N e. V /\ K e. V ) /\ N =/= K /\ E. e e. ( Edg ` G ) { K , N } C_ e ) )
4		simp1l	|- ( ( ( N e. V /\ K e. V ) /\ N =/= K /\ E. e e. ( Edg ` G ) { K , N } C_ e ) -> N e. V )
5	3 4	sylbi	|- ( N e. ( G NeighbVtx K ) -> N e. V )