The set $\{1, 2\}$ is a proper subset of $\{1, 2, 3\}$. But $\{1, 2, 3\}$ itself is not.
More generally, we might want to define a notion of "proper-ness" that derives from this basic notion of proper super- and subsets. For example, I might want to talk about a sequence $(f_i)_{i = 1}^n$ of surjective functions, and call it "proper" if not one of the $f_i$s is a bijection. Then I might have some way of manipulating such sequences, and I might want to write that "... is not in general preserved under [such a manipulation]", where ... is the name of the state of being proper.
What do we call the property of being proper? Do we call it "propriety", perhaps, or something else?