Definition:Greatest Set by Set Inclusion

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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 the subset relation.

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

That is:

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

Class Theory

Let $A$ be a class.

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

$(1): \quad M \in A$
$(2): \quad \forall S: \paren {S \in A \implies S \subseteq M}$

Also see