Metamath Proof Explorer


Theorem usgr2edg1

Description: If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 8-Jun-2021)

Ref Expression
Hypotheses usgrf1oedg.i
|- I = ( iEdg ` G )
usgrf1oedg.e
|- E = ( Edg ` G )
Assertion usgr2edg1
|- ( ( ( G e. USGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> -. E! x e. dom I N e. ( I ` x ) )

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 umgr2edg1
 |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> -. E! x e. dom I N e. ( I ` x ) )
5 3 4 sylanl1
 |-  ( ( ( G e. USGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> -. E! x e. dom I N e. ( I ` x ) )