Theorem nbgrisvtx

Description: Every neighbor N of a vertex K is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 26-Oct-2020) (Revised by AV, 12-Feb-2022)

	Ref	Expression
Hypothesis	nbgrisvtx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	nbgrisvtx	⊢ ( 𝑁 ∈ ( 𝐺 NeighbVtx 𝐾 ) → 𝑁 ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		nbgrisvtx.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
3	1 2	nbgrel	⊢ ( 𝑁 ∈ ( 𝐺 NeighbVtx 𝐾 ) ↔ ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ) ∧ 𝑁 ≠ 𝐾 ∧ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝐾 , 𝑁 } ⊆ 𝑒 ) )
4		simp1l	⊢ ( ( ( 𝑁 ∈ 𝑉 ∧ 𝐾 ∈ 𝑉 ) ∧ 𝑁 ≠ 𝐾 ∧ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) { 𝐾 , 𝑁 } ⊆ 𝑒 ) → 𝑁 ∈ 𝑉 )
5	3 4	sylbi	⊢ ( 𝑁 ∈ ( 𝐺 NeighbVtx 𝐾 ) → 𝑁 ∈ 𝑉 )