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 ) |
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 ) |