Metamath Proof Explorer

Theorem vtxnbuvtx

Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017) (Revised by AV, 30-Oct-2020) (Proof shortened by AV, 14-Feb-2022)

Ref Expression
Hypothesis uvtxel.v ⊢ V = Vtx ⁡ G
Assertion vtxnbuvtx ⊢ N ∈ UnivVtx ⁡ G → ∀ n ∈ V ∖ N n ∈ G NeighbVtx N

Proof

Step Hyp Ref Expression
1 uvtxel.v ⊢ V = Vtx ⁡ G
2 1 uvtxel ⊢ N ∈ UnivVtx ⁡ G ↔ N ∈ V ∧ ∀ n ∈ V ∖ N n ∈ G NeighbVtx N
3 2 simprbi ⊢ N ∈ UnivVtx ⁡ G → ∀ n ∈ V ∖ N n ∈ G NeighbVtx N