Metamath Proof Explorer

Theorem usgr2edg

Description: If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Feb-2021)

Ref Expression
Hypotheses usgrf1oedg.i ⊢ I = iEdg ⁡ G
usgrf1oedg.e ⊢ E = Edg ⁡ G
Assertion usgr2edg ⊢ G ∈ USGraph ∧ A ≠ B ∧ N A ∈ E ∧ B N ∈ E → ∃ x ∈ dom ⁡ I ∃ y ∈ dom ⁡ I x ≠ y ∧ N ∈ I ⁡ x ∧ N ∈ I ⁡ y

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 umgr2edg ⊢ G ∈ UMGraph ∧ A ≠ B ∧ N A ∈ E ∧ B N ∈ E → ∃ x ∈ dom ⁡ I ∃ y ∈ dom ⁡ I x ≠ y ∧ N ∈ I ⁡ x ∧ N ∈ I ⁡ y
5 3 4 sylanl1 ⊢ G ∈ USGraph ∧ A ≠ B ∧ N A ∈ E ∧ B N ∈ E → ∃ x ∈ dom ⁡ I ∃ y ∈ dom ⁡ I x ≠ y ∧ N ∈ I ⁡ x ∧ N ∈ I ⁡ y