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 ) )
5 4 adantr 
 |-  ( ( ( 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 ) ) )
8 7 ad2ant2r 
 |-  ( ( ( 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 ) ) )
11 10 ad2ant2rl 
 |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( B e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) )
12 11 simpld 
 |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> B e. ( Vtx ` G ) )
13 8 simpld 
 |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> N e. ( Vtx ` G ) )
14 simpr 
 |-  ( ( ( G e. UMGraph /\ A =/= B ) /\ ( { N , A } e. E /\ { B , N } e. E ) ) -> ( { N , A } e. E /\ { B , N } e. E ) )
15 1 2 6 uhgr2edg 
 |-  ( ( ( 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 ) ) )
16 5 9 12 13 14 15 syl131anc 
 |-  ( ( ( 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 ) ) )