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 ( Paths ` G ) P <-> ( F ( SPaths ` G ) P \/ F ( Cycles ` G ) P ) )

Proof

Step	Hyp	Ref	Expression
1		pthisspthorcycl	|- ( F ( Paths ` G ) P -> ( F ( SPaths ` G ) P \/ F ( Cycles ` G ) P ) )
2		spthispth	|- ( F ( SPaths ` G ) P -> F ( Paths ` G ) P )
3		cyclispth	|- ( F ( Cycles ` G ) P -> F ( Paths ` G ) P )
4	2 3	jaoi	|- ( ( F ( SPaths ` G ) P \/ F ( Cycles ` G ) P ) -> F ( Paths ` G ) P )
5	1 4	impbii	|- ( F ( Paths ` G ) P <-> ( F ( SPaths ` G ) P \/ F ( Cycles ` G ) P ) )