probability - Expected value of the negative part of a random variable

I want to prove that, if $X$ is a real valued random variable with finite expected value, then:

$$\mathbb{E}[X]=\displaystyle \int_{0}^{\infty} \mathbf{P}(X \geq t)dt - \int_{-\infty}^{0} \mathbf{P} (X \leq t)dt.$$

We have that $$\mathbb{E}[X] = \mathbb {E} (X^{+}) - \mathbb{E} (X^{-})$$
and I know how to prove that if $Y$ is a non-negative r.v., then its expected value can be expressed as $$\mathbb{E}[Y]=\displaystyle \int_{0}^{\infty} \mathbf{P}(Y \geq t)dt.$$

I am having trouble expressing the second integral as the expectation of the negative part, $X^{-}$.

Can anyone help me with that?