# Doubleton of Elements is Subset

## Theorem

Let $S$ be a set.

Let $\set {x,y}$ be the doubleton of distinct $x$ and $y$.

Then:

$x, y \in S \iff \set {x,y} \subseteq S$

## Proof

### Necessary Condition

Let $x, y \in S$.

$\set x \subseteq S$
$\set y \subseteq S$
$\set x \cup \set y \subseteq S$
$\set x \cup \set y = \set {x, y}$

Hence:

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

$\Box$

### Sufficient Condition

Let $\set {x,y} \subseteq S$.

From the definition of a subset:

$x \in \set {x,y} \implies x \in S$
$y \in \set {x,y} \implies y \in S$

$\blacksquare$