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.(adsbygoogle = window.adsbygoogle || []).push({});

Also, suppose we adopt the anti-foundation axiom instead, why is there no set of all non-well-founded sets?

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# Proper Classes in ZF

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