Theorem pthspthcyc

Description: A pair <. F , P >. represents a path if it represents either a simple path or a cycle. The exclusivity only holds for non-trivial paths ( F =/= (/) ), see cyclnspth . (Contributed by AV, 2-Oct-2025)

	Ref	Expression
Assertion	pthspthcyc	$$⊢F\mathrm{Paths}\left(G\right)P\leftrightarrow \left(F\mathrm{SPaths}\left(G\right)P\vee F\mathrm{Cycles}\left(G\right)P\right)$$

Proof

Step	Hyp	Ref	Expression
1		pthisspthorcycl	$$⊢F\mathrm{Paths}\left(G\right)P\to \left(F\mathrm{SPaths}\left(G\right)P\vee F\mathrm{Cycles}\left(G\right)P\right)$$
2		spthispth	$$⊢F\mathrm{SPaths}\left(G\right)P\to F\mathrm{Paths}\left(G\right)P$$
3		cyclispth	$$⊢F\mathrm{Cycles}\left(G\right)P\to F\mathrm{Paths}\left(G\right)P$$
4	2 3	jaoi	$$⊢\left(F\mathrm{SPaths}\left(G\right)P\vee F\mathrm{Cycles}\left(G\right)P\right)\to F\mathrm{Paths}\left(G\right)P$$
5	1 4	impbii	$$⊢F\mathrm{Paths}\left(G\right)P\leftrightarrow \left(F\mathrm{SPaths}\left(G\right)P\vee F\mathrm{Cycles}\left(G\right)P\right)$$