Definition:Greatest Set by Set Inclusion
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Definition
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:
- $(1): \quad M \in A$
- $(2): \quad \forall S: \paren {S \in A \implies S \subseteq M}$