Description: The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wspthsswwlkn | |- ( N WSPathsN G ) C_ ( N WWalksN G ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wspthnp | |- ( w e. ( N WSPathsN G ) -> ( ( N e. NN0 /\ G e. _V ) /\ w e. ( N WWalksN G ) /\ E. f f ( SPaths ` G ) w ) ) |
|
| 2 | 1 | simp2d | |- ( w e. ( N WSPathsN G ) -> w e. ( N WWalksN G ) ) |
| 3 | 2 | ssriv | |- ( N WSPathsN G ) C_ ( N WWalksN G ) |