Metamath Proof Explorer

Theorem wspthnfi

Description: In a finite graph, the set of simple paths of a fixed length is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018) (Revised by AV, 18-May-2021)

Ref Expression
Assertion wspthnfi ⊢ Vtx ⁡ G ∈ Fin → N WSPathsN G ∈ Fin

Proof

Step Hyp Ref Expression
1 wwlksnfi ⊢ Vtx ⁡ G ∈ Fin → N WWalksN G ∈ Fin
2 wspthsswwlkn ⊢ N WSPathsN G ⊆ N WWalksN G
3 2 a1i ⊢ Vtx ⁡ G ∈ Fin → N WSPathsN G ⊆ N WWalksN G
4 1 3 ssfid ⊢ Vtx ⁡ G ∈ Fin → N WSPathsN G ∈ Fin