Theorem clnbgrn0

Description: The closed neighborhood of a vertex is never empty. (Contributed by AV, 16-May-2025)

	Ref	Expression
Hypothesis	clnbgrn0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	clnbgrn0	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 ClNeighbVtx 𝑁 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		clnbgrn0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	clnbgrvtxel	⊢ ( 𝑁 ∈ 𝑉 → 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝑁 ) )
3		ne0i	⊢ ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝑁 ) → ( 𝐺 ClNeighbVtx 𝑁 ) ≠ ∅ )
4	2 3	syl	⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 ClNeighbVtx 𝑁 ) ≠ ∅ )