Example of an interesting theorem that fails in intuitionistic set theory but is classically valid?

Question

I'm interested in intuitionistic set theories at the moment. I know that lots of principles imply LEM and so fail intuitionistically, and also a few basic principles - linear ordering of ordinals, for example - fail in intuitionistic set theories. However, lots of classically valid theorems - Cantor's springs to mind - nevertheless go through. Are there any interesting combinatorial bits of set theory that don't work intuitonistically?

Answer 1

Here is a very nice example from MathOverflow (see this thread, I particularly like Andrej Bauer's well written answer).

The Cantor-Bernstein Theorem. If there is an injective function $f\colon A\to B$ and there is an injective function $g\colon B\to A$, then there is a bijection function $h\colon A\to B$.