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