In Bonn, we've been have a discussion on the topic in the title:



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> Suppose that $A$ and $B$ are classes and that there are injections from $A$ to $B$ and from $B$ to $A$. Does it follow that there is a bijection between $A$ and $B$?
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Example: Let $A$ the class of sets of cardinality one and let $B$ be the class of sets of cardinality two. There is an injection


$A\to B$ sending $a$ to $\{a,\varnothing\}$,


$B\to A$ sending b to $\{\{b\}\}$.


Does it follow that there is a bijection between $A$ and $B$?



Does Cantor-Bernstein hold for classes?
Ignoring set-theoretic technicalities of formulating the question properly, I see no reason that the usual proof of Schroder-Bernstein wouldn't work.


(Set-theoretic technicalities: In the standard language of set theory, you can't quantify over classes, so you can't quite state this. However, you can prove a metatheorem saying that whenever you exhibit two such injections, you can prove there is also a bijection. Alternatively, you could work in [set theory with classes](http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory), in which the statement can be made properly and you ought to be able to prove it just like ordinary Schroder-Bernstein. Alternatively, it is a trivial corollary of the "global" axiom of choice (which implies, in particular, that all proper classes have the same size), though this is kind of applying a sledgehammer.)


