Metamath Proof Explorer

Theorem uspgrloopvd2

Description: The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop ), the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015) (Revised by Alexander van der Vekens, 22-Dec-2017) (Revised by AV, 17-Dec-2020) (Proof shortened by AV, 21-Feb-2021)

Ref Expression
Hypothesis uspgrloopvtx.g ⊢ G = V A N
Assertion uspgrloopvd2 ⊢ V ∈ W ∧ A ∈ X ∧ N ∈ V → VtxDeg ⁡ G ⁡ N = 2

Proof

Step Hyp Ref Expression
1 uspgrloopvtx.g ⊢ G = V A N
2 1 uspgrloopvtx ⊢ V ∈ W → Vtx ⁡ G = V
3 2 3ad2ant1 ⊢ V ∈ W ∧ A ∈ X ∧ N ∈ V → Vtx ⁡ G = V
4 simp2 ⊢ V ∈ W ∧ A ∈ X ∧ N ∈ V → A ∈ X
5 simp3 ⊢ V ∈ W ∧ A ∈ X ∧ N ∈ V → N ∈ V
6 1 uspgrloopiedg ⊢ V ∈ W ∧ A ∈ X → iEdg ⁡ G = A N
7 6 3adant3 ⊢ V ∈ W ∧ A ∈ X ∧ N ∈ V → iEdg ⁡ G = A N
8 3 4 5 7 1loopgrvd2 ⊢ V ∈ W ∧ A ∈ X ∧ N ∈ V → VtxDeg ⁡ G ⁡ N = 2