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 $S$.
$\blacksquare$