Let $A$ be some set of axis-aligned rectangles in the plane, each pair of which has empty intersection. Prove that $A$ is a countable set.
(An axis-aligned rectangle is a set of the form
$$M = {\{\langle x,y \rangle \in \mathbb{R^2} | a \leq x \leq b , c \leq y \leq d}\}$$ for $a,b,c,d$ such that $a < b$ and $ c < d$.)
Attempt:
I tried using the density of the $\mathbb{Q}$ in $(\mathbb{R},\leq)$, but without any success.