Smallest Set is Unique

Theorem

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.

Then the smallest set of $\TT$, if it exists, must be unique.

Proof

Let $A, B \in \TT$ both be smallest sets of $\TT$.

Since $A$ is the smallest set:

$A \subseteq B$

Since $B$ is the smallest set:

$B \subseteq A$

Hence, by definition of set equality:

$A = B$

Therefore the smallest set of $\TT$ is unique.

$\blacksquare$