When proving in class that REGtm or EMPTYtm are undecidable , we formalized a reduction from ATM to REGtm or EMPTYtm.
The main idea was to assume there is a TM R that decides REGtm or EMPTYtm and using R to construct a TM that decides ATM and get a contradiction
The logic structure we learned in recitation for building a reduction was to come up with a good reduction from ATM to EMPTYtm and because ATM is undecidable, so is EMPTYtm
My question is how does the contradiction helps us in this case ? I mean that I don't understand the combination of contradiction and reduction in the same proof- is'nt one of these two enough for proving decidability ?