A subset$A$ of a set$S$ is proper if it is not the improper subset of $S$ ($S$ as a subset of itself). We may also say (in the context of concrete sets) that $S$ is a proper superset of $A$ or that $S$ properly contains?$A$.

Definition

In material set theory

In material set theory, we may state that $A$ is proper in any of these equivalent ways:

$A$ is not equal (as a subset) to $S$;

There exists an element $x$ of $S$ such that $x \notin A$;

In structural set theory, subsets are represented by injections, and so we may state that the injection $m:A \hookrightarrow S$ is proper in any of these equivalent ways:

Similarly as the case in material set theory, these three definitions are equivalent only if we accept the principle of excluded middle.

More generally

In any category $C$, subsets become subobjects and are thus represented by the monomorphisms in $C$. Then a proper subobject is defined in any of the following ways:

If $C$ is a well-pointed category with terminal object$1$, then there exists a global element $x:1 \to S$ such that for all global elements $y:1 \to A$, $m \circ x \neq y$;