Metamath Proof Explorer

Theorem usgr2edg1

Description: If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 8-Jun-2021)

Ref Expression
Hypotheses usgrf1oedg.i ⊢ I = iEdg ⁡ G
usgrf1oedg.e ⊢ E = Edg ⁡ G
Assertion usgr2edg1 ⊢ G ∈ USGraph ∧ A ≠ B ∧ N A ∈ E ∧ B N ∈ E → ¬ ∃! x ∈ dom ⁡ I N ∈ I ⁡ x

Proof

Step Hyp Ref Expression
1 usgrf1oedg.i ⊢ I = iEdg ⁡ G
2 usgrf1oedg.e ⊢ E = Edg ⁡ G
3 usgrumgr ⊢ G ∈ USGraph → G ∈ UMGraph
4 1 2 umgr2edg1 ⊢ G ∈ UMGraph ∧ A ≠ B ∧ N A ∈ E ∧ B N ∈ E → ¬ ∃! x ∈ dom ⁡ I N ∈ I ⁡ x
5 3 4 sylanl1 ⊢ G ∈ USGraph ∧ A ≠ B ∧ N A ∈ E ∧ B N ∈ E → ¬ ∃! x ∈ dom ⁡ I N ∈ I ⁡ x