How do I show this Mobius transformation maps the unit disk onto itself?

Question

I am struggling to show that, for any $\alpha, z \in \mathbb C$ such that $|\alpha|<1$ and $|z|<1$:

$$\left | \frac{z - \alpha}{1-\bar\alpha z} \right | < 1$$

This should be doable with simple algebric manipulations, without using the properties of Mobius transformations, but I cannot figure out how.

Thank you for your time.