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# State/Axiom of power sets

Axiom: (Axiom of Power Sets.) Suppose that $x$ is a set. There exists a set $y$, such that: $$\forall u, u \in y \Leftrightarrow u \subset x.$$

In other words, there exists a set, whose elements are precisely the subsets of $x$.

This is one of the Zermelo-Fraenkel axioms of set theory.

## Logical Connections

This statement logically relies on the following definitions and statements:

The following statements and definitions rely on the material of this page: Def/Cartesian product, Def/Power set, and State/Surjective functions have right inverses

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