Theorem sclnbgrisvtx

Description: Every member X of the semiclosed neighborhood of a vertex N is a vertex. (Contributed by AV, 16-May-2025)

	Ref	Expression
Hypotheses	dfsclnbgr2.v	|- V = ( Vtx ` G )
	dfsclnbgr2.s	|- S = { n e. V | E. e e. E { N , n } C_ e }
	dfsclnbgr2.e	|- E = ( Edg ` G )
Assertion	sclnbgrisvtx	|- ( X e. S -> X e. V )

Proof

Step	Hyp	Ref	Expression
1		dfsclnbgr2.v	|- V = ( Vtx ` G )
2		dfsclnbgr2.s	|- S = { n e. V | E. e e. E { N , n } C_ e }
3		dfsclnbgr2.e	|- E = ( Edg ` G )
4	1 2 3	sclnbgrel	|- ( X e. S <-> ( X e. V /\ E. e e. E { N , X } C_ e ) )
5	4	simplbi	|- ( X e. S -> X e. V )