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I have been asked to decide whether the set {$x: \exists y (y \in x)$} exists.

This translates as the set of all non-empty sets. Does this exist? I know that there is no set of all sets but how do I prove there is no set of non-empty sets? Do I use the union?

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This does not exist. If it did, then you could take its union with $\{\emptyset\}$ to get the set of all sets, which you already know is impossible.

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