Description: In a simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018) (Revised by AV, 5-Jun-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | usgr2pthspth | |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( Paths ` G ) P <-> F ( SPaths ` G ) P ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pthistrl | |- ( F ( Paths ` G ) P -> F ( Trails ` G ) P ) |
|
| 2 | usgr2trlspth | |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( Trails ` G ) P <-> F ( SPaths ` G ) P ) ) |
|
| 3 | 1 2 | syl5ib | |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( Paths ` G ) P -> F ( SPaths ` G ) P ) ) |
| 4 | spthispth | |- ( F ( SPaths ` G ) P -> F ( Paths ` G ) P ) |
|
| 5 | 3 4 | impbid1 | |- ( ( G e. USGraph /\ ( # ` F ) = 2 ) -> ( F ( Paths ` G ) P <-> F ( SPaths ` G ) P ) ) |