Theorem clnbgrcl

Description: If a class X has at least one element in its closed neighborhood, this class must be a vertex. (Contributed by AV, 7-May-2025)

	Ref	Expression
Hypothesis	clnbgrcl.v	|- V = ( Vtx ` G )
Assertion	clnbgrcl	|- ( N e. ( G ClNeighbVtx X ) -> X e. V )

Proof

Step	Hyp	Ref	Expression
1		clnbgrcl.v	|- V = ( Vtx ` G )
2		df-clnbgr	|- ClNeighbVtx = ( g e. _V , v e. ( Vtx ` g ) |-> ( { v } u. { n e. ( Vtx ` g ) | E. e e. ( Edg ` g ) { v , n } C_ e } ) )
3	2	mpoxeldm	|- ( N e. ( G ClNeighbVtx X ) -> ( G e. _V /\ X e. [_ G / g ]_ ( Vtx ` g ) ) )
4		csbfv	|- [_ G / g ]_ ( Vtx ` g ) = ( Vtx ` G )
5	4 1	eqtr4i	|- [_ G / g ]_ ( Vtx ` g ) = V
6	5	eleq2i	|- ( X e. [_ G / g ]_ ( Vtx ` g ) <-> X e. V )
7	6	biimpi	|- ( X e. [_ G / g ]_ ( Vtx ` g ) -> X e. V )
8	3 7	simpl2im	|- ( N e. ( G ClNeighbVtx X ) -> X e. V )