nLab
proper subset

Idea

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

We may state that $A$ is proper in any of these equivalent ways:

1. $A$ is not equal (as a subset) to $S$;
2. There exists an element $x$ of $S$ such that $x \notin A$;
3. Given any way of expressing $A$ as the intersection of a family of subsets of $S$, this family is inhabited.

Actually, these three definitions are equivalent only if we accept the principle of excluded middle; in constructive mathematics, we usually prefer (3). (For example, consider the notion of proper filter on a set $X$, thought of as a subset of the power set of $X$.) However, (3) is not predicative; see positive element for discussion of this in the dual context. Also, (2) may be strengthened using an inequality relation other than the denial inequality.