\begin{definition}[Definition:Non-Empty Set]
Let $S$ be a set.
Then $S$ is said to be '''non-empty''' {{iff}} $S$ has at least one element.
By the Axiom of Extension, this may also be phrased as:
:$S \ne \O$
where $\O$ denotes the empty set.
Many mathematical theorems and definitions require sets to be '''non-empty''' in order to avoid erratic results and inconsistencies.
\end{definition}