Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure

Theorem
Let $\mathcal B$ be a set of subsets of $\R$, the set of all real numbers.

Let
 * $\left\vert{\mathcal B}\right\vert < \mathfrak c$

where
 * $\left\vert{\mathcal B}\right\vert$ denotes the cardinality of $\mathcal B$,
 * $\mathfrak c = \left\vert{\R}\right\vert$ denotes continuum.

Then
 * $\exists x, y \in \R: x < y \land \left[{x \,.\,.\, y}\right) \notin \left\{{\bigcup \mathcal G: \mathcal G \subseteq \mathcal B}\right\}$