# Intersection of Doubleton

## Theorem

Let $x$ and $y$ be sets.

Let $\set {x, y}$ be a doubleton.

Then $\ds \bigcap \set {x, y}$ is a set such that:

$\ds \bigcap \set {x, y} = x \cap y$

## Proof

 $\ds$  $\ds z \in \bigcap \set {x, y}$ $\ds \leadstoandfrom \ \$ $\ds$  $\ds \forall w \in \set {x, y}: z \in w$ Definition of Intersection of Class $\ds \leadstoandfrom \ \$ $\ds$  $\ds \forall w: \paren {\paren {w = x \land w = y} \land z \in w}$ Definition of Doubleton Class $\ds \leadstoandfrom \ \$ $\ds$  $\ds \paren {z \in x \land z \in y}$ Equality implies Substitution $\ds \leadstoandfrom \ \$ $\ds$  $\ds z \in \paren {x \cap y}$ Definition of Class Intersection

From Intersection of Non-Empty Class is Set it follows that $x \cap y$ is a set.

$\blacksquare$