Existence and Uniqueness of Monotone Class 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 $\mathfrak m \left({\mathcal G}\right)$, the monotone class generated by $\mathcal G$, exists and is unique.

Existence
By Power Set is Monotone Class, there is at least one monotone class containing $\mathcal G$.

Now let $\Bbb M$ be the collection of monotone classes containing $\mathcal G$:


 * $\Bbb M := \left\{{\mathfrak{m}': \mathcal G \subseteq \mathfrak{m}', \text{$\mathfrak{m}'$ is a monotone class}}\right\}$

By Intersection of Monotone Classes is Monotone Class, $\mathfrak m := \bigcap \Bbb M$ is a monotone class.

Also, by Set Intersection Preserves Subsets:
 * $\mathcal G \subseteq \mathfrak m$

Now let $\mathfrak{m}'$ be a monotone class containing $\mathcal G$.

Then by construction of $\mathfrak m$, and Intersection is Subset: General Result:
 * $\mathfrak m \subseteq \mathfrak{m}'$

Uniqueness
Suppose $\mathfrak{m}_1$ and $\mathfrak{m}_2$ are both monotone classes generated by $\mathcal G$.

Then property $(2)$ for these monotone classes implies both $\mathfrak{m}_1 \subseteq \mathfrak{m}_2$ and $\mathfrak{m}_2 \subseteq \mathfrak{m}_1$.

By definition of set equality:
 * $\mathfrak{m}_1 = \mathfrak{m}_2$