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 $\set 0 \not \subseteq \set 1$:

$\set 0$ is not the smallest set of $\TT$.

Similarly, since $\set 1 \not \subseteq \set 0$:

$\set 1$ is not the smallest set of $\TT$.

Therefore $\TT$ has no smallest set.

$\blacksquare$