Generated Sigma-Algebra Contains Generated Dynkin System

Theorem
Let $X$ be a set.

Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Then $\delta \left({\mathcal G}\right) \subseteq \sigma \left({\mathcal G}\right)$.

Here $\delta$ denotes generated Dynkin system, and $\sigma$ denotes generated $\sigma$-algebra.

Proof
By Sigma-Algebra is Dynkin System, $\sigma \left({\mathcal G}\right)$ is a Dynkin system.

The definition of $\delta \left({\mathcal G}\right)$ now ensures that $\delta \left({\mathcal G}\right) \subseteq \sigma \left({\mathcal G}\right)$.