# 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

$\O \in \powerset {\set x}$

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

That is:

 $\displaystyle$  $\displaystyle A \subseteq \set x \land A \ne \O$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle A \subseteq \set x \land \exists y : y \in A$ Definition of Empty Set $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle A \subseteq \set x \land \exists y : y \in A \land y \in \set x$ Definition of Subset $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle A \subseteq \set x \land \exists y : y \in A \land y = x$ Definition of Singleton $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle A \subseteq \set x \land x \in A$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle A \subseteq \set x \land \set x \subseteq A$ Singleton of Element is Subset $\displaystyle \leadsto \ \$ $\displaystyle$  $\displaystyle A = \set x$ Definition of Set Equality

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

$\blacksquare$