Power Set of Doubleton

Theorem

Let $x, y$ be distinct objects.

Then the power set of the doubleton $\set {x, y}$ is:

$\powerset {\set {x, y} } = \set {\O, \set x, \set y, \set {x, y} }$

Proof

By definition of a subset:

$\set x , \set y, \set {x, y} \subseteq \set {x, y}$

Let $A \subseteq \set {x, y}$:

$A \ne \set x, \set y, \set {x, y}$

From set equality:

$\set {x, y} \nsubseteq A$

From Doubleton of Elements is Subset:

either $x \notin A$ or $y \notin A$.

Without loss of generality assume that $x \notin A$.

From Intersection With Singleton is Disjoint if Not Element:

$A \cap \set x = \O$

From Subset of Set Difference iff Disjoint Set:

$A \subseteq \set {x, y} \setminus \set x$

From Set Difference of Doubleton and Singleton is Singleton:

$A \subseteq \set y$

From set equality:

$\set y \nsubseteq A$

From Singleton of Element is Subset:

$y \notin A$.

From Intersection With Singleton is Disjoint if Not Element:

$A \cap \set y = \O$

From Subset of Set Difference iff Disjoint Set:

$A \subseteq \set y \setminus \set y$

From Set Difference with Self is Empty Set:

$A \subseteq \O$

From Empty Set is Subset of All Sets:

$\O \subseteq A$

From set equality:

$A = \O$

It follows that:

$\powerset {\set {x, y} } = \set {\O, \set x, \set y, \set {x, y} }$

$\blacksquare$