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Metamath Proof Explorer

Theorem wspthsswwlknon

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 between the two vertices. (Contributed by AV, 15-May-2021)

Ref Expression
Assertion wspthsswwlknon 
|- ( A ( N WSPathsNOn G ) B ) C_ ( A ( N WWalksNOn G ) B )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 1 wspthnonp 
 |-  ( w e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( w e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A ( SPathsOn ` G ) B ) w ) ) )
3 simp3l 
 |-  ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. ( Vtx ` G ) /\ B e. ( Vtx ` G ) ) /\ ( w e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A ( SPathsOn ` G ) B ) w ) ) -> w e. ( A ( N WWalksNOn G ) B ) )
4 2 3 syl 
 |-  ( w e. ( A ( N WSPathsNOn G ) B ) -> w e. ( A ( N WWalksNOn G ) B ) )
5 4 ssriv 
 |-  ( A ( N WSPathsNOn G ) B ) C_ ( A ( N WWalksNOn G ) B )