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 = ( Vtx ‘ 𝐺 )
Assertion	clnbgrisvtx	⊢ ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) → 𝑁 ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		clnbgrvtxel.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
3	1 2	clnbgrel	⊢ ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) ↔ ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ) ∧ ( 𝑁 = 𝐾 ∨ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝐾 , 𝑁 } ⊆ 𝑒 ) ) )
4		simpll	⊢ ( ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ) ∧ ( 𝑁 = 𝐾 ∨ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝐾 , 𝑁 } ⊆ 𝑒 ) ) → 𝑁 ∈ 𝑉 )
5	3 4	sylbi	⊢ ( 𝑁 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) → 𝑁 ∈ 𝑉 )