Prove that there's no decidable language that separates two other languages.

Question

I was reviewing for an exam and I found this question: Let A and B be two disjoint languages (that is, A ∩ B = ∅). Say that a language C separates A and B iff A ⊆ C and B ⊆ (not C) . Define two disjoint languages by

A = {⟨M,w⟩: M is a TM and M accepts w}

B = {⟨M,w⟩: M isaTMandM rejects w}

Prove that there does not exist any decidable language C that separates A and B.

I guess the way the proof has to be made is by contradiction. If we were able to built a decidable language C that separates A and B then we would be able to use that decision as a black box to decide A={<M,w>| M is a TM and M accepts w}, that we know that's undecidable, and this would be a contradiction. But could someone elaborate this reduction? I have been thinking on it for days. Thanks

Answer 1

If $C$ is decidable, then the questions ''$x\in A$'' or ''$x\in B$'' must be decidable since $C$ has the separation property. But these questions are not decidable due to the halting problem.