Theorem clnbgrisvtx

Description: Every member N of the closed neighborhood of a vertex K is a vertex. (Contributed by AV, 9-May-2025)

	Ref	Expression
Hypothesis	clnbgrvtxel.v	|- V = ( Vtx ` G )
Assertion	clnbgrisvtx	|- ( N e. ( G ClNeighbVtx K ) -> N e. V )

Proof

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	|- V = ( Vtx ` G )
2		eqid	|- ( Edg ` G ) = ( Edg ` G )
3	1 2	clnbgrel	|- ( N e. ( G ClNeighbVtx K ) <-> ( ( N e. V /\ K e. V ) /\ ( N = K \/ E. e e. ( Edg ` G ) { K , N } C_ e ) ) )
4		simpll	|- ( ( ( 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 ClNeighbVtx K ) -> N e. V )