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	Could not format assertion : No typesetting found for |- ( K e. V -> K e. ( G ClNeighbVtx K ) ) with typecode |-

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	Could not format ( K e. ( G ClNeighbVtx K ) <-> ( ( K e. V /\ K e. V ) /\ ( K = K \/ E. e e. ( Edg ` G ) { K , K } C_ e ) ) ) : No typesetting found for |- ( K e. ( G ClNeighbVtx K ) <-> ( ( K e. V /\ K e. V ) /\ ( K = K \/ E. e e. ( Edg ` G ) { K , K } C_ e ) ) ) with typecode |-
7	2 2 4 6	syl21anbrc	Could not format ( K e. V -> K e. ( G ClNeighbVtx K ) ) : No typesetting found for |- ( K e. V -> K e. ( G ClNeighbVtx K ) ) with typecode |-