Metamath Proof Explorer


Theorem usgredg

Description: For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017) (Revised by AV, 17-Oct-2020) (Shortened by AV, 25-Nov-2020.)

Ref Expression
Hypotheses edgssv2.v V = Vtx G
edgssv2.e E = Edg G
Assertion usgredg G USGraph C E a V b V a b C = a b

Proof

Step Hyp Ref Expression
1 edgssv2.v V = Vtx G
2 edgssv2.e E = Edg G
3 usgrumgr G USGraph G UMGraph
4 1 2 umgredg G UMGraph C E a V b V a b C = a b
5 3 4 sylan G USGraph C E a V b V a b C = a b