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 )