Unit of System of Sets is Unique

Theorem
The unit of a system of sets, if it exists, is unique.

If $U$ is the unit of a system of sets $\mathcal S$, then $\forall A \in \mathcal S: A \subseteq U$.

Proof
Let $\mathcal S$ be a system of sets.

Suppose $U$ and $U'$ are both units of $\mathcal S$.

Then, by definition:
 * $\forall A \in \mathcal S: A \cap U = A$
 * $\forall A \in \mathcal S: A \cap U' = A$

This applies to both $U$ and $U'$, of course.

So $U \cap U' = U$ and $U' \cap U = U'$.

From Intersection with Subset is Subset‎ it follows that $U \subseteq U'$ and $U' \subseteq U$.

By definition of set equality:
 * $U = U'$

We also see that from Intersection with Subset is Subset, $A \cap U = A \iff A \subseteq U$, which shows that:
 * $\forall A \in \mathcal S: A \subseteq U$.