# Metamath Proof Explorer

## Theorem usgr2edg

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 ) ) )`

### Proof

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 ) ) )`