Inadmissible theorems in research
Does maths research have anything inadmissible?


No, but trying to prove X without using Y is still a very useful concept even in research, because it can lead to interesting generalizations, or new proof techniques that can be applied to a larger set of problems. 

For instance, in some sense the Lebesgue integral is "just" trying to prove the properties of integrals without using the continuity of f, or the theory of matroids is "just" trying to prove the properties of linearly independent vectors without using a lot of properties from the vector space structure.

So this is far from being a pointless exercise, if that's what you had in mind.