Smallest Set may not Exist
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Theorem
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.
The smallest set of $\TT$ may not exist.
Proof
Let $S = \set {0, 1}$ and $\TT = \set {\set 0, \set 1} \in \powerset S$.
Then since $\set 0 \nsubseteq \set 1$:
- $\set 0$ is not the smallest set of $\TT$.
Similarly, since $\set 1 \nsubseteq \set 0$:
- $\set 1$ is not the smallest set of $\TT$.
Therefore $\TT$ has no smallest set.
$\blacksquare$