Exists Subset which is not Element

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Theorem

Let $S$ be a set.

Then there exists at least one subset of $S$ which is not an element of $S$.

Proof 1

Let $S$ be a set.

Let $T$ be the set of all elements of $S$ which do not contain $S$ as elements.

Then by the corollary to Russell's paradox $T$ itself cannot be an element of $S$.

This $T$ is the required subset.

Proof 2

Consider the power set $\powerset S$ of $S$.

From Cantor's Theorem, there is no surjection $f: S \to \powerset S$.

That is, there are more subsets of $S$ than there are elements of $S$.

So there must be at least one subset of $S$ which is not an element of $S$.

$\blacksquare$