Theorem: Given some cardinal K, the is no set of all sets equinumerous to K.
I have been thinking about this for a few days and can't come up with a proof. Intuitively, it seems to me such a set could be used to construct the set of all sets which is not allowed. This would be obvious if there is a set of all cardinals. But I don't think this needs to be relied upon. (Because my book certainly has not given me the tools to prove that statement either way.)
Either complete proofs or hints would be appreciated.