Equivalent Conditions for Cover by Collection of Subsets

Theorem
Let $X$ be a set.

Then the following conditions are equivalent for a subset $\mathcal C \subseteq \mathcal P \left({X}\right)$ of the power set of $X$:
 * $\left({1}\right): \quad$ $\mathcal C$ is a cover for $X$.
 * $\left({2}\right): \quad$ $\displaystyle X = \bigcup \mathcal C$.
 * $\left({3}\right): \quad$ $\displaystyle \exists \mathcal S \subseteq \mathcal C: X = \bigcup \mathcal S$.