# inequality of distances in a graph

Certainly it’s obvious but I can’t catch the reason behind it.

Why do we have :

Let $$D= (V,A)$$ be a directed graph, $$w:A \to \mathbb R$$ be arc weights and $$s \in V$$. Denote with $$d(s,v)$$ the length of the shortest path from $$s$$ to $$v$$ in $$D$$, subject to $$w$$.

If there are no negative cycles in $$D$$, then we have $$\forall (u,v) \in A : d(s,v) \leq d(s,u) + w(u,v) \iff d(s,v)- d(s,u) \leq w(u,v).$$

Why?