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For any f and A, we can define X as { a in A | a is not in f(a) } . That set exists in ZFC by the axiom of separation, also known as the axiom of subsets or the axiom of comprehension.

I recommend you pick up a book on ZFC if you are interested in understanding set theory. I found Enderton’s “Elements of Set Theory” to be a really good introductory text.