Power Set Exists and is Unique

Theorem
Let $V$ be a basic universe.

Let $x \in V$ be a set.

Let $\powerset x$ denote the power set of $x$.

Then $\powerset x$ is guaranteed to exist and is unique.

Proof
By Axiom of Powers, $\powerset x$ exists and is a set.

Let $\powerset x$, $\map \QQ x$ both be power sets of $x$.

From definition of power sets, $\forall T$:
 * $T \in \powerset x \iff T \subseteq x$
 * $T \in \map \QQ x \iff T \subseteq x$

Since Biconditional is Commutative and Transitive:
 * $T \in \powerset x \iff T \in \map \QQ x$

By Axiom of Extension, $\powerset x = \map \QQ x$.

Hence the power set is unique.