# Set Difference of Doubleton and Singleton is Singleton

## Theorem

Let $x, y$ be distinct objects.

Then:

$\set{x, y} \setminus \set x = \set y$

## Proof

 $\ds \set{x, y} \setminus \set x$ $=$ $\ds \big \{ z: z \in \set{x, y} \land z \notin \set x \big \}$ Definition of Set Difference $\ds$ $=$ $\ds \big \{ z: \paren{ z = x \lor z = y} \land z \notin \set x \big \}$ Definition of Doubleton $\ds$ $=$ $\ds \big \{ z: \paren{ z = x \lor z = y} \land z \ne x \big \}$ Definition of Singleton $\ds$ $=$ $\ds \big \{ z: \paren{z = x \land z \ne x} \lor \paren {z = y \land z \ne x} \big \}$ Conjunction Distributes over Disjunction $\ds$ $=$ $\ds \big \{ z: \bot \lor \paren {z = y \land z \ne x} \big \}$ Definition of Contradiction $\ds$ $=$ $\ds \big \{ z: \paren {z = y \land z \ne x} \big \}$ Disjunction with Contradiction $\ds$ $=$ $\ds \big \{ z : z = y \big \}$ Rule of Simplification $\ds$ $=$ $\ds \set y$ Definition of Singleton

$\blacksquare$