Metamath Proof Explorer

Theorem wwlksnon0

Description: Sufficient conditions for a set of walks of a fixed length between two vertices to be empty. (Contributed by AV, 15-May-2021) (Proof shortened by AV, 21-May-2021)

		Ref	Expression
	Hypothesis	wwlksnon0.v	$⊢ V = Vtx (G)$
	Assertion	wwlksnon0	$⊢ \neg ((N \in ℕ_{0} \land G \in V) \land (A \in V \land B \in V)) \to A (N WWalksNOn G) B = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		wwlksnon0.v	$⊢ V = Vtx (G)$
2		df-wwlksnon	$⊢ WWalksNOn = (n \in ℕ_{0}, g \in V ⟼ (a \in Vtx (g), b \in Vtx (g) ⟼ \{w \in (n WWalksN g) \| (w (0) = a \land w (n) = b)\}))$
3	1	wwlksnon	$⊢ (N \in ℕ_{0} \land G \in V) \to N WWalksNOn G = (a \in V, b \in V ⟼ \{w \in (N WWalksN G) \| (w (0) = a \land w (N) = b)\})$
4	2 3	2mpo0	$⊢ \neg ((N \in ℕ_{0} \land G \in V) \land (A \in V \land B \in V)) \to A (N WWalksNOn G) B = \emptyset$