Sigma-Algebra Contains Generated Sigma-Algebra of Subset

Theorem
Let $\sigma_{\mathcal F}$ be a  be a $\sigma$-algebra on a set $\mathcal F$.

Let $\sigma_{\mathcal F}$ contain a set of sets $\mathcal E$.

Let $\sigma \left({\mathcal E}\right)$ be the $\sigma$-algebra generated by $\mathcal E$.

Then $\sigma \left({\mathcal E}\right) \subseteq \sigma_{\mathcal F}$

Proof
$\sigma_{\mathcal F}$ is a $\sigma$-algebra containing $\mathcal E$.

$\sigma \left({\mathcal E}\right)$ is a subset of all $\sigma$-algebras containing $\mathcal F$, by definition of a generated $\sigma$-algebra.

Therefore it contains $\sigma \left({\mathcal E}\right)$.