# Greatest Set is Unique

## Theorem

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.

## Proof

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.

$\blacksquare$