I know that V, Ord and Card are proper classes because otherwise foundation, successor and Cantor's theorem would be violated respectively. But if a class is in bijection with one of them, why is that class automatically proper? If we don't assume choice, then the cardinality argument doesn't work. So let's not assume choice. Also, suppose we adopt the anti-foundation axiom instead, why is there no set of all non-well-founded sets?