Definition:Greatest Set by Set Inclusion/Class Theory

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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 known as

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

Also see