Dynkin System with Generator Closed under Intersection is Sigma-Algebra

Theorem
Let $X$ be a set.

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

Suppose that $\mathcal G$ satisfies the following condition:


 * $(1):\quad \forall G, H \in \mathcal G: G \cap H \in \mathcal G$

That is, $\mathcal G$ is closed under intersection.

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

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