We may state that is proper in any of these equivalent ways:
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 , thought of as a subset of the power set of .) 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.