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If I understand correctly (which is far from guaranteed, so a reply telling me that it is rubbish will be better than no reply at all):
Suppose we have two models N and M such that N is a (non-conservative) extension of M, such that the class S is a set under N but not under M, with respect to which S is a proper class. Then we can talk about the cardinality of S (having it understood that we are working under N). So, if we have such a class S, and another class T which is a set under N such that |S|=|T|, then T is also a proper class with respect to M.
Questions: Correct? If not, please point out where and why. If so, why? Thanks in advance.

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    $\begingroup$ What is meant by "class" in this context? In $\mathsf{ZFC}$ and related systems, classes are definable (with parameters). Is this the intention here? (By the way, "large cardinal" is a technical term. I've removed the incorrect tag.) $\endgroup$ Dec 18, 2013 at 7:23
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    $\begingroup$ You probably need some additional assumptions: Otherwise, it could be that $M$ is countable in $N$, which immediately gives a negative answer to your question. Perhaps we want cardinals of $M$ to be cardinals in $N$? (It is not enough to simply ask that $N$ is an end-extension to prevent this situation, by the way.) $\endgroup$ Dec 18, 2013 at 7:35

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