Smallest Set may not Exist

Theorem
Let $S$ be a set.

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

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

The smallest set of $\TT$ may not exist.

Proof
Let $S = \set {0, 1}$ and $\TT = \set {\set 0, \set 1} \in \powerset S$.

Then since $\set 0 \not \subseteq \set 1$:
 * $\set 0$ is not the smallest set of $\TT$.

Similarly, since $\set 1 \not \subseteq \set 0$:
 * $\set 1$ is not the smallest set of $\TT$.

Therefore $\TT$ has no smallest set.