Power Set of Singleton

Theorem
Let $x$ be an object.

Then the power set of the singleton $\set x$ is:
 * $\powerset {\set x} = \set {\O, \set x}$

Proof
From Empty Set is Subset of All Sets:
 * $\O \in \powerset {\set x}$

Let $A \in \powerset {\set x}$ such that $A \ne \O$

That is:

So a subset of $\set x$ is either $\O$ or $\set x$.