Theorem clnbgrssvtx

Description: The closed neighborhood of a vertex K in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025)

	Ref	Expression
Hypothesis	clnbgrvtxel.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
Assertion	clnbgrssvtx	$$⊢GClNeighbVtxK\subseteq V$$

Proof

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2	1	clnbgrisvtx	$$⊢n\in \left(GClNeighbVtxK\right)\to n\in V$$
3	2	ssriv	$$⊢GClNeighbVtxK\subseteq V$$