# Metamath Proof Explorer

## Theorem umgr2edg

Description: If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017) (Revised by AV, 11-Feb-2021)

Ref Expression
Hypotheses usgrf1oedg.i
`|- I = ( iEdg ` G )`
usgrf1oedg.e
`|- E = ( Edg ` G )`
Assertion 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 ) ) )`

### Proof

Step Hyp Ref Expression
1 usgrf1oedg.i
` |-  I = ( iEdg ` G )`
2 usgrf1oedg.e
` |-  E = ( Edg ` G )`
3 umgruhgr
` |-  ( G e. UMGraph -> G e. UHGraph )`
4 3 anim1i
` |-  ( ( G e. UMGraph /\ A =/= B ) -> ( G e. UHGraph /\ A =/= B ) )`
` |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( G e. UHGraph /\ A =/= B ) )`
6 eqid
` |-  ( Vtx ` G ) = ( Vtx ` G )`
7 6 2 umgrpredgv
` |-  ( ( G e. UMGraph /\ { N , A } e. E ) -> ( N e. ( Vtx ` G ) /\ A e. ( Vtx ` G ) ) )`
` |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( N e. ( Vtx ` G ) /\ A e. ( Vtx ` G ) ) )`
9 8 simprd
` |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> A e. ( Vtx ` G ) )`
10 6 2 umgrpredgv
` |-  ( ( G e. UMGraph /\ { B , N } e. E ) -> ( B e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) )`
` |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( B e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) )`
` |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> B e. ( Vtx ` G ) )`
` |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> N e. ( Vtx ` G ) )`
` |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( { N , A } e. E /\ { B , N } e. E ) )`
` |-  ( ( ( G e. UHGraph /\ A =/= B ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) /\ ( { 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 ) ) )`
` |-  ( ( ( 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 ) ) )`