spthonisspth

Description: A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018) (Revised by AV, 18-Jan-2021)

		Ref	Expression
	Assertion	spthonisspth	$⊢ F (A SPathsOn (G) B) P \to F SPaths (G) P$

Proof

Step	Hyp	Ref	Expression
1		eqid	$⊢ Vtx (G) = Vtx (G)$
2	1	spthonprop	$⊢ F (A SPathsOn (G) B) P \to ((G \in V \land A \in Vtx (G) \land B \in Vtx (G)) \land (F \in V \land P \in V) \land (F (A TrailsOn (G) B) P \land F SPaths (G) P))$
3		simp3r	$⊢ ((G \in V \land A \in Vtx (G) \land B \in Vtx (G)) \land (F \in V \land P \in V) \land (F (A TrailsOn (G) B) P \land F SPaths (G) P)) \to F SPaths (G) P$
4	2 3	syl	$⊢ F (A SPathsOn (G) B) P \to F SPaths (G) P$

Metamath Proof Explorer

Theorem spthonisspth

Proof