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	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to N\mathrm{WSPathsN}G\in \mathrm{Fin}$$

Proof

Step	Hyp	Ref	Expression
1		wwlksnfi	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to N\mathrm{WWalksN}G\in \mathrm{Fin}$$
2		wspthsswwlkn	$$⊢N\mathrm{WSPathsN}G\subseteq N\mathrm{WWalksN}G$$
3	2	a1i	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to N\mathrm{WSPathsN}G\subseteq N\mathrm{WWalksN}G$$
4	1 3	ssfid	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{Fin}\to N\mathrm{WSPathsN}G\in \mathrm{Fin}$$