Theorem clnbgrvtxel

Description: Every vertex K is a member of its closed neighborhood. (Contributed by AV, 10-May-2025)

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

Proof

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		id	$$⊢K\in V\to K\in V$$
3		eqidd	$$⊢K\in V\to K=K$$
4	3	orcd	$$⊢K\in V\to \left(K=K\vee \exists e\in \mathrm{Edg}\left(G\right)\left\{K,K\right\}\subseteq e\right)$$
5		eqid	$$⊢\mathrm{Edg}\left(G\right)=\mathrm{Edg}\left(G\right)$$
6	1 5	clnbgrel	$$⊢K\in \left(GClNeighbVtxK\right)\leftrightarrow \left(\left(K\in V\wedge K\in V\right)\wedge \left(K=K\vee \exists e\in \mathrm{Edg}\left(G\right)\left\{K,K\right\}\subseteq e\right)\right)$$
7	2 2 4 6	syl21anbrc	$$⊢K\in V\to K\in \left(GClNeighbVtxK\right)$$