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	$$⊢V=\mathrm{Vtx}\left(G\right)$$
Assertion	clnbgrcl	$$⊢N\in \left(GClNeighbVtxX\right)\to X\in V$$

Proof

Step	Hyp	Ref	Expression
1		clnbgrcl.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		df-clnbgr	$$⊢ClNeighbVtx=\left(g\in \mathrm{V},v\in \mathrm{Vtx}\left(g\right)⟼\left\{v\right\}\cup \left\{n\in \mathrm{Vtx}\left(g\right)|\exists e\in \mathrm{Edg}\left(g\right)\left\{v,n\right\}\subseteq e\right\}\right)$$
3	2	mpoxeldm	$$⊢N\in \left(GClNeighbVtxX\right)\to \left(G\in \mathrm{V}\wedge X\in ⦋G/g⦌\mathrm{Vtx}\left(g\right)\right)$$
4		csbfv	$$⊢⦋G/g⦌\mathrm{Vtx}\left(g\right)=\mathrm{Vtx}\left(G\right)$$
5	4 1	eqtr4i	$$⊢⦋G/g⦌\mathrm{Vtx}\left(g\right)=V$$
6	5	eleq2i	$$⊢X\in ⦋G/g⦌\mathrm{Vtx}\left(g\right)\leftrightarrow X\in V$$
7	6	biimpi	$$⊢X\in ⦋G/g⦌\mathrm{Vtx}\left(g\right)\to X\in V$$
8	3 7	simpl2im	$$⊢N\in \left(GClNeighbVtxX\right)\to X\in V$$