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Assume the following set:

$\{X|X\notin A\}$

That is the set of all sets that is not an element of the set $A$. How can I prove this set does not exist?

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Since the set $\mathcal{P}(A)$ of all subsets of $A$ exists, if your set existed, then their union would exist, too. But this union is the set of all sets, which doesn't exist.

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