Empty Set from Principle of Non-Contradiction

Theorem
The empty set can be characterised as:


 * $\O := \set {x: x \in E \text { and } x \notin E}$

where $E$ is an arbitrary set.

Proof
$x \in \O$ as defined here.

Thus we have:
 * $x \in E$

and:
 * $x \notin E$

This is a contradiction.

It follows by Proof by Contradiction that $x \notin \O$.

Hence, as $x$ was arbitrary, there can be no $x$ such that $x \in \O$.

Thus $\O$ is the empty set by definition.