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 Subset Equivalences it follows that $$U \subseteq U'$$ and $$U' \subseteq U$$.

From Equality of Sets, it follows that $$U = U'$$.

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