Generated Sigma-Algebra Contains Generated Sigma-Algebra of Subset

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

Let $\sigma\left({\mathcal F}\right)$ contain a set of sets $\mathcal E$.

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

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

Proof
Apply Sigma-Algebra Contains Generated Sigma-Algebra of Subset to $\sigma\left({\mathcal F}\right)$.