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 ) ) |
| 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 ) ) |