FOM: Cantor's theorem of little interest in constructive math

[Ayan]

There can be one use of these cardinality theorem in mathematics, say, I 
want to prove that two groups are non-isomorphic and
I settle the question as that the groups have different cardinality, I can 
think of a proof like this but I don't know how important it is? Is it 
possible to bypass the argument by showing a direct contradiction. Probably 
it will be too case dependent to discuss generally.

But one place I know cardinality is definitely in use (correct me if I am 
wrong) is that there are more Lebesgue measurable sets than Borel 
measurable using the completeness of L-measure and the cantor set. Does 
anybody know of any argument to bypass the obvious use of cardinality here.

--Ayan