Description: There is no empty closed walk (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018) (Revised by AV, 24-Apr-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | clwwlkgt0 | |- ( W e. ( ClWWalks ` G ) -> 0 < ( # ` W ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid | |- ( Vtx ` G ) = ( Vtx ` G ) |
|
| 2 | 1 | clwwlkbp | |- ( W e. ( ClWWalks ` G ) -> ( G e. _V /\ W e. Word ( Vtx ` G ) /\ W =/= (/) ) ) |
| 3 | hashgt0 | |- ( ( W e. Word ( Vtx ` G ) /\ W =/= (/) ) -> 0 < ( # ` W ) ) |
|
| 4 | 3 | 3adant1 | |- ( ( G e. _V /\ W e. Word ( Vtx ` G ) /\ W =/= (/) ) -> 0 < ( # ` W ) ) |
| 5 | 2 4 | syl | |- ( W e. ( ClWWalks ` G ) -> 0 < ( # ` W ) ) |