Metamath Proof Explorer

Theorem usgrpredgv

Description: An edge of a simple graph always connects two vertices. Analogue of usgredgprv . (Contributed by Alexander van der Vekens, 7-Oct-2017) (Revised by AV, 9-Jan-2020) (Revised by AV, 23-Oct-2020) (Proof shortened by AV, 27-Nov-2020)

Ref Expression
Hypotheses usgredgppr.e ⊢ E = Edg ⁡ G
usgrpredgv.v ⊢ V = Vtx ⁡ G
Assertion usgrpredgv ⊢ G ∈ USGraph ∧ M N ∈ E → M ∈ V ∧ N ∈ V

Proof

Step Hyp Ref Expression
1 usgredgppr.e ⊢ E = Edg ⁡ G
2 usgrpredgv.v ⊢ V = Vtx ⁡ G
3 usgrumgr ⊢ G ∈ USGraph → G ∈ UMGraph
4 2 1 umgrpredgv ⊢ G ∈ UMGraph ∧ M N ∈ E → M ∈ V ∧ N ∈ V
5 3 4 sylan ⊢ G ∈ USGraph ∧ M N ∈ E → M ∈ V ∧ N ∈ V