Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself ([Dedekind infinite](https://en.wikipedia.org/wiki/Dedekind-infinite_set)). Without Choice it is consistent that there are infinite sets which can't be partitioned into two infinite subsets ([amorphous sets](https://en.wikipedia.org/wiki/Amorphous_set)), so the above statement no longer holds since a bijection to a proper subset implies a partition into two disjoint infinite subsets as proven on the wiki -- all of this is discussed in the question and answers [here](https://mathoverflow.net/questions/86654/what-sort-of-structure-can-amorphous-sets-support) much more succinctly.



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> Is it consistent in $MK$ without Global Choice that there are amorphous proper classes, meaning proper classes which can't be partitioned into two proper class sized subclasses? 
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Directly generalizing the argument given on the wiki article for amorphous sets seems to require a notion of [transfinite function composition](https://ncatlab.org/nlab/show/transfinite+composition) which can be defined in good categorical generality using colimits, but it is not immediately apparent how to generalize the recursive definition of the $S\_i$'s for limit ordinal $i$ since the given definitions depend on immediate predecessor steps.



Amorphous proper classes in MK
Unless I'm missing something, the answer is **no**: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and this lets us partition the original class into two proper classes, for example $s^{-1}(limits)$ versus $s^{-1}(successors)$.


