Definition:Smallest Set by Set Inclusion

Definition

Let $S$ be a set.

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

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

Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by the subset relation.

Then $T \in \TT$ is the smallest set of $\TT$ if and only if $T$ is the smallest element of $\struct {\TT, \subseteq}$.

That is:

$\forall X \in \TT: T \subseteq X$

Let $A$ be a class.

Then a set $m$ is the smallest element of $A$ (with respect to the subset relation) if and only if:

$(1): \quad m \in A$

$(2): \quad \forall S: \paren {S \in A \implies m \subseteq S}$