Theorem clnbgrn0

Description: The closed neighborhood of a vertex is never empty. (Contributed by AV, 16-May-2025)

	Ref	Expression
Hypothesis	clnbgrn0.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
Assertion	clnbgrn0	$$⊢N\in V\to GClNeighbVtxN\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		clnbgrn0.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2	1	clnbgrvtxel	$$⊢N\in V\to N\in \left(GClNeighbVtxN\right)$$
3		ne0i	$$⊢N\in \left(GClNeighbVtxN\right)\to GClNeighbVtxN\ne \varnothing$$
4	2 3	syl	$$⊢N\in V\to GClNeighbVtxN\ne \varnothing$$