Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum

Theorem
Let $\mathcal B$ be a set of subsets of $\R$.

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.

Let
 * $X = \left\{{x \in \R: \exists U \in \left\{{\bigcup B: B \subseteq \mathcal B}\right\}: x}\right.$ is local minimum in $\left.{U}\right\}$.

Then:
 * $\left\vert{X}\right\vert < \mathfrak c$.