Theorem clnbgrisvtx

Description: Every member N of the closed neighborhood of a vertex K is a vertex. (Contributed by AV, 9-May-2025)

	Ref	Expression
Hypothesis	clnbgrvtxel.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
Assertion	clnbgrisvtx	Could not format assertion : No typesetting found for |- ( N e. ( G ClNeighbVtx K ) -> N e. V ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		eqid	$$⊢\mathrm{Edg}\left(G\right)=\mathrm{Edg}\left(G\right)$$
3	1 2	clnbgrel	Could not format ( N e. ( G ClNeighbVtx K ) <-> ( ( N e. V /\ K e. V ) /\ ( N = K \/ E. e e. ( Edg ` G ) { K , N } C_ e ) ) ) : No typesetting found for |- ( N e. ( G ClNeighbVtx K ) <-> ( ( N e. V /\ K e. V ) /\ ( N = K \/ E. e e. ( Edg ` G ) { K , N } C_ e ) ) ) with typecode |-
4		simpll	$$⊢\left(\left(N\in V\wedge K\in V\right)\wedge \left(N=K\vee \exists e\in \mathrm{Edg}\left(G\right)\left\{K,N\right\}\subseteq e\right)\right)\to N\in V$$
5	3 4	sylbi	Could not format ( N e. ( G ClNeighbVtx K ) -> N e. V ) : No typesetting found for |- ( N e. ( G ClNeighbVtx K ) -> N e. V ) with typecode |-