complex analysis - Inequality involving Mobius transformation

I have the following simple-looking inequality I have to show:

Let $z, w \in \mathbb D$, where $\mathbb D$ is the open unit disc in $\mathbb C$. Show that
$$\left| \frac{z-w}{1-\overline{z}w} \right| \geq \left| \frac{|z|-|w|}{1-|z||w|} \right|.$$

It looks pretty straightforward, but I just can't seem to get it, and I think I might be missing something obvious. I've tried putting $z=|z|e^{i \alpha}$ and $w=|w|e^{i \beta}$ to get
$$\left| \frac{z-w}{1-\overline{z}w} \right| = \left| \frac{|z|-|w|e^{i \theta}}{1-|z||w|e^{i \theta}} \right|$$
where $\theta = \beta - \alpha$, and can't get much out of this. I've tried squaring both sides etc., and a few other things. If anyone has any ideas, I'd be very grateful, thanks.