Reply #13: You need to clarify the question. [View All]

[Jim__]

If m is any set, then there exists a set u such that for all x, x is an element of u if and only if there exists y such that x is an element of y and y is an element of m.

This sounds very much like an axiom of Zermelo-Fraenkel, the Axiom schema of specification, with u = y.

Outside of Zermelo-Fraenkel, the claim is not always true. For instance, in a set theory that allows for urelements, let m be a set all of whose elements are urelements. Then, the claim that x is an element of y is meaningless when y is an element of m.