The axioms we have introduced so far do not ensure the existence of a power set for a set $X$, containing all the subsets of $X$ as its elements. For instance, the axiom of separation ensures the existence of any subset of $X$ separately, and we could use the axiom of pairing to create a set containing any two of such subsets as elements, but it is not possible to combine all subsets of a given set at once. For this reason, we need another axiom, the axiom of power set.
For each set \(X\) there exists a set containing all subsets of $X$, formally:
$$\forall X~\exists~Y~\forall z~(z\in Y\Rightarrow z\subseteq X).$$
Corollaries: 1
Axioms: 1
Definitions: 2 3
Proofs: 4