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 e. USGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )

Proof

Step Hyp Ref Expression
1 usgredgppr.e
 |-  E = ( Edg ` G )
2 usgrpredgv.v
 |-  V = ( Vtx ` G )
3 usgrumgr
 |-  ( G e. USGraph -> G e. UMGraph )
4 2 1 umgrpredgv
 |-  ( ( G e. UMGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )
5 3 4 sylan
 |-  ( ( G e. USGraph /\ { M , N } e. E ) -> ( M e. V /\ N e. V ) )