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Well, it's not that difficult to fix it in ZF.

If you define S to be S = {x in U: x not in x}, then it simply means S is not in U.

There is no U in ZF so I really don't understand what you are saying.

You can call U anything you want. It's by axiom schema of specification: for any set A, there some set B with a set C in B iff C in A.