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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 $\subseteq$ considered as an ordering.

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

That is:

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

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