Definition:Smallest Set by Set Inclusion

Definition
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathcal T \subseteq \mathcal P \left({S}\right)$ be a subset of $\mathcal P \left({S}\right)$.

Let $\left({\mathcal T, \subseteq}\right)$ be the ordered set formed on $\mathcal T$ by $\subseteq$ considered as an ordering.

Then $T \in \mathcal T$ is the smallest set of $\mathcal T$ iff $T$ is the smallest element of $\left({\mathcal T, \subseteq}\right)$.

That is:
 * $\forall X \in \mathcal T: T \subseteq X$