Description: If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Feb-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | usgrf1oedg.i | |- I = ( iEdg ` G ) |
|
| usgrf1oedg.e | |- E = ( Edg ` G ) |
||
| Assertion | usgr2edg | |- ( ( ( G e. USGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> E. x e. dom I E. y e. dom I ( x =/= y /\ N e. ( I ` x ) /\ N e. ( I ` y ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | usgrf1oedg.i | |- I = ( iEdg ` G ) |
|
| 2 | usgrf1oedg.e | |- E = ( Edg ` G ) |
|
| 3 | usgrumgr | |- ( G e. USGraph -> G e. UMGraph ) |
|
| 4 | 1 2 | umgr2edg | |- ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> E. x e. dom I E. y e. dom I ( x =/= y /\ N e. ( I ` x ) /\ N e. ( I ` y ) ) ) |
| 5 | 3 4 | sylanl1 | |- ( ( ( G e. USGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> E. x e. dom I E. y e. dom I ( x =/= y /\ N e. ( I ` x ) /\ N e. ( I ` y ) ) ) |