  It is well-known that Choice and Regularity are independent of each other but
have important common consequences of logical character (reflection principles,
representations of classes by sets, etc.). We explain this phenomenon by
isolating their "intersection", a principle (called here Best-Foundedness)
which is consistent with the negations of both axioms but implies all these
consequences. Then we study relationships between these consequences (and near
principles) in detail. Finally, we consider some arguments related to truth of
various principles in set theory, especially arguments concerning the
interpretability strength.
