\begin{definition}[Definition:Proper Subset]
If a set $S$ is a subset of another set $T$, that is, $S \subseteq T$, and also:
:$S \ne T$
:$S \ne \O$
then $S$ is referred to as a '''proper subset''' of $T$.
The set $T$ '''properly contains''', or '''strictly contains''', the set $S$.
If $S \subseteq T$ and $S \ne T$, then the notation $S \subsetneqq T$ is used.
If we wish to refer to a set which we specifically require not to be empty, we can denote it like this:
:$\O \subsetneqq S$
and one which we want to specify as possibly being empty, we write:
:$\O \subseteq S$
Thus for $S$ to be a '''proper subset''' of $T$, we can write it as $\O \subsetneqq S \subsetneqq T$.
\end{definition}