Definition:Proper Subset

Definition
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 $\O \subsetneqq S \subsetneqq T$.