Set of Pairwise Disjoint Intervals is Countable

Theorem
Let $X$ be a subset of $\mathcal P \left({\R}\right)$ such that
 * $X$ is mutually disjoint:
 * $\forall A,B \in X: A \ne B \implies A \cap B = \varnothing$.
 * $X$ contains open intervals only:
 * $\forall A \in X: \exists x, y \in \R: x < y \land A = \left({x \,.\,.\, y}\right)$.

Then $X$ is countable.