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Metamath Proof Explorer

Theorem wspthsswwlknon

Description: The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length between the two vertices. (Contributed by AV, 15-May-2021)

Ref Expression
Assertion wspthsswwlknon ⊢ A N WSPathsNOn G B ⊆ A N WWalksNOn G B

Proof

Step Hyp Ref Expression
1 eqid ⊢ Vtx ⁡ G = Vtx ⁡ G
2 1 wspthnonp ⊢ w ∈ A N WSPathsNOn G B → N ∈ ℕ 0 ∧ G ∈ V ∧ A ∈ Vtx ⁡ G ∧ B ∈ Vtx ⁡ G ∧ w ∈ A N WWalksNOn G B ∧ ∃ f f A SPathsOn ⁡ G B w
3 simp3l ⊢ N ∈ ℕ 0 ∧ G ∈ V ∧ A ∈ Vtx ⁡ G ∧ B ∈ Vtx ⁡ G ∧ w ∈ A N WWalksNOn G B ∧ ∃ f f A SPathsOn ⁡ G B w → w ∈ A N WWalksNOn G B
4 2 3 syl ⊢ w ∈ A N WSPathsNOn G B → w ∈ A N WWalksNOn G B
5 4 ssriv ⊢ A N WSPathsNOn G B ⊆ A N WWalksNOn G B