# Axiom:Axiom of Powers

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## Axiom

### Set Theory

For every set, there exists a set of sets whose elements are all the subsets of the given set.

- $\forall x: \exists y: \paren {\forall z: \paren {z \in y \iff \paren {w \in z \implies w \in x} } }$

### Class Theory

Let $x$ be a set.

Then its power set $\powerset x$ is also a set.

## Also known as

The **axiom of powers** is also known as the **axiom of the power set**.