Theorem nbgrssvtx

Description: The neighbors of a vertex K in a graph form a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 26-Oct-2020) (Revised by AV, 12-Feb-2022)

	Ref	Expression
Hypothesis	nbgrisvtx.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
Assertion	nbgrssvtx	$$⊢G\mathrm{NeighbVtx}K\subseteq V$$

Proof

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