Existence and Uniqueness of Monotone Class Generated by Collection of Subsets

Theorem
Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.

Then $\map {\mathfrak m} \GG$, the monotone class generated by $\GG$, exists and is unique.

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

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


 * $\Bbb M := \set {\mathfrak m': \GG \subseteq \mathfrak m', \mathfrak m' \text{ is a monotone class} }$

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

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

Now let $\mathfrak m'$ be a monotone class containing $\GG$.

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 $\GG$.

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:
 * $\mathfrak m_1 = \mathfrak m_2$