nLab
proper subset

A subset $A$ of a set $S$ is proper if it is not the improper subset of $S$ ($S$ as a subset of itself). In fact, we may state that $A$ is proper in either or these equivalent ways:

• $A$ is not equal (as a subset) to $S$;
• There exists an element $x$ of $S$ such that $x\notin A$;
• 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 the last definition. (For example, consider the notion of proper filter on a set $X$, thought of as a subset of the power set of $X$.) However, this last definition is not predicative; see positive element for discussion of this in the dual context.