In mathematics, the

**axiom of power set**is one of the Zermelo-Fraenkel axioms of axiomatic set theory.

In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:

- ∀
*A*, ∃*B*, ∀*C*,*C*∈*B*↔ (∀*D*,*D*∈*C*→*D*∈*A*);

- Given any set
*A*, there is a set*B*such that, given any set*C*,*C*is a member of*B*if and only if, given any set*D*, if*D*is a member of*C*, then*D*is a member of*A*.

*C*is a subset of

*A*. Thus, what the axiom is really saying is that, given a set

*A*, we can find a set

*B*whose members are precisely the subsets of

*A*. We can use the axiom of extensionality to show that this set

*B*is unique. We call the set

*B*the

*power set*of

*A*, and denote it

**P**

*A*. Thus the essence of the axiom is:

- Every set has a power set.