Existence and Uniqueness of Dynkin System Generated by Collection of Subsets

Theorem
Let $X$ be a set.

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

Then $\delta \left({\mathcal G}\right)$, the Dynkin system generated by $\mathcal G$, exists and is unique.

Existence
By Power Set is Dynkin System, there exists at least one Dynkin system containing $\mathcal G$.

Next, let $\Bbb D$ be the collection of Dynkin systems containing $\mathcal G$:


 * $\Bbb D := \left\{{\mathcal{D}': \mathcal G \subseteq \mathcal{D}', \text{$\mathcal{D}'$ is a Dynkin system}}\right\}$

By Intersection of Dynkin Systems is Dynkin System, $\mathcal D := \bigcap \Bbb D$ is a Dynkin system.

Also, by Set Intersection Preserves Subsets: General Case, $\mathcal G \subseteq \mathcal D$.

Now let $\mathcal{D}'$ be a Dynkin system containing $\mathcal G$.

By construction of $\mathcal D$, and Intersection is Subset: General Result, $\mathcal D \subseteq \mathcal{D}'$.

Uniqueness
Suppose both $\mathcal{D}_1$ and $\mathcal{D}_2$ are Dynkin systems generated by $\mathcal G$.

Then property $(2)$ for these Dynkin systems implies both $\mathcal{D}_1 \subseteq \mathcal{D}_2$ and $\mathcal{D}_2 \subseteq \mathcal{D}_1$.

By definition of set equality:
 * $\mathcal{D}_1 = \mathcal{D}_2$