# Unit of System of Sets is Unique

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## Theorem

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

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

## Proof

Let $\SS$ be a system of sets.

Suppose $U$ and $U'$ are both units of $\SS$.

Then, by definition:

- $\forall A \in \SS: A \cap U = A$
- $\forall A \in \SS: A \cap U' = A$

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

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

From Intersection with Subset is Subsetâ€Ž 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:

- $\forall A \in \SS: A \subseteq U$.

$\blacksquare$