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Let $S$ be a set.

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

Let $\mathcal T \subseteq \powerset S$ be a subset of $\powerset S$.

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

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

That is:

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

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