The empty set is the set $\text{Ø}$ defined by
\[ \text{Ø}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ x\neq x\right\} , \]
where $X$ is the set in the set existence axiom, 
of .
