Metamath Proof Explorer


Theorem nbgrisvtx

Description: Every neighbor N of a vertex K is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017) (Revised by AV, 26-Oct-2020) (Revised by AV, 12-Feb-2022)

Ref Expression
Hypothesis nbgrisvtx.v
|- V = ( Vtx ` G )
Assertion nbgrisvtx
|- ( N e. ( G NeighbVtx K ) -> N e. V )

Proof

Step Hyp Ref Expression
1 nbgrisvtx.v
 |-  V = ( Vtx ` G )
2 eqid
 |-  ( Edg ` G ) = ( Edg ` G )
3 1 2 nbgrel
 |-  ( N e. ( G NeighbVtx K ) <-> ( ( N e. V /\ K e. V ) /\ N =/= K /\ E. e e. ( Edg ` G ) { K , N } C_ e ) )
4 simp1l
 |-  ( ( ( N e. V /\ K e. V ) /\ N =/= K /\ E. e e. ( Edg ` G ) { K , N } C_ e ) -> N e. V )
5 3 4 sylbi
 |-  ( N e. ( G NeighbVtx K ) -> N e. V )