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

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

assume that $x \notin A$.

Then:

By definition of subset:
 * $A \subseteq \set y$

By definition of the power set:
 * $A \in \powerset {\set y}$

From Power Set of Singleton:
 * $\powerset {\set y} = \big \{ \O, \set y \big \}$

Since $A \ne \set y$, then:
 * $A = \O$

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