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

Aiming for a contradiction, suppose $x \in \O$ 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.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems