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?