Metamath Proof Explorer


Theorem vtxdumgr0nedg

Description: If a vertex in a multigraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017) (Revised by AV, 12-Dec-2020) (Proof shortened by AV, 15-Dec-2020)

Ref Expression
Hypotheses vtxdushgrfvedg.v V=VtxG
vtxdushgrfvedg.e E=EdgG
vtxdushgrfvedg.d D=VtxDegG
Assertion vtxdumgr0nedg GUMGraphUVDU=0¬vVUvE

Proof

Step Hyp Ref Expression
1 vtxdushgrfvedg.v V=VtxG
2 vtxdushgrfvedg.e E=EdgG
3 vtxdushgrfvedg.d D=VtxDegG
4 umgruhgr GUMGraphGUHGraph
5 1 2 3 vtxduhgr0nedg GUHGraphUVDU=0¬vVUvE
6 4 5 syl3an1 GUMGraphUVDU=0¬vVUvE