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.