Countability of disjoint positive-length intervals

A family of pairwise disjoint real intervals of positive length is countable. Each interval contains a rational number in its interior by the density of the rational numbers. Fix an enumeration of the rational numbers and select the first member lying inside each interval. Disjointness makes this selection an injection into a countable set. The argument also applies to disjoint nonempty open subsets of the real line.