Power Set of Doubleton

Theorem
Let $x, y$ be distinct objects.

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

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:
 * $\set {x,y} \not \subseteq A$

From Doubleton of Elements is Subset:
 * either $x \notin A$ or $y \notin A$.

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:
 * $\set y \not \subseteq A$

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:
 * $\powerset {\set {x, y}} = \big \{ \O, \set x, \set y, \set {x,y} \big \}$