Greatest Set is Unique
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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 $\powerset S$.
Then the greatest set of $\TT$, if it exists, must be unique.
Let $A, B \in \TT$ both be greatest sets of $\TT$.
Since $A$ is the greatest set:
- $B \subseteq A$
Since $B$ is the greatest set:
- $A \subseteq B$
Hence, by definition of set equality:
- $A = B$
Therefore the greatest set of $\TT$ is unique.