Power Set of Singleton

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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:

\(\displaystyle \) \(\) \(\displaystyle A \subseteq \set x \land A \ne \O\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \) \(\) \(\displaystyle A \subseteq \set x \land \exists y : y \in A\) 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 $\set x$.

$\blacksquare$