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 ) |
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 ) |