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Let $\AA$ be a set of sets or a class of sets.

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

$M \in \AA$ and
$\forall S: \paren {S \in \AA \implies S \subseteq M}$

Also known as

Some sources use the term largest set. The two terms are synonymous.

Also see


2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.): Definition $3.4.8$