Smallest Set is Unique
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.
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:
- $A = B$