Consider two random variables $X$ and $Y$. $X$ is said to precede $Y$ in first order stochastic dominance sense if $\mathbb{E}[v(X) ] \leq \mathbb{E}[v(Y) ]$ for every non-decreasing function $v: \mathbb{R} \to \mathbb{R}$ or, equivalently, if $ \forall x \in \mathbb{R}: \mathbb{P}\left\{X \leq x\right\}\geq \mathbb{P}\left\{Y \leq x\right\}. $
I understand how the first definition implies the second one (choose $v:\mathbb{R} \to \mathbb{R}: z \mapsto I_{]x, +\infty[}(z) $ for any $ x \in \mathbb{R}$ fixed). However, I can't seem to figure out why they are equivalent.
Thanks in advance.