Cardinality of Union not greater than Product

Theorem
Let $\mathcal F$ be a set of sets.

Let:
 * $\left\vert{\mathcal F}\right\vert \leq \mathbf m$

where
 * $\left\vert{\mathcal F}\right\vert$ denotes the cardinality of $\mathcal F$,
 * $\mathbf m$ is cardinal number (possibly infinite).

Let:
 * $\forall A \in \mathcal F: \left\vert{A}\right\vert \leq \mathbf n$

where
 * $\mathbf n$ is cardinal number (possibly infinite).

Then:
 * $\displaystyle \left\vert{\bigcup \mathcal F}\right\vert \leq \left\vert{\mathbf m \times \mathbf n}\right\vert = \mathbf m \mathbf n$