The problem:

Show that the set $\displaystyle \{\neg(p\rightarrow q), \neg(q\rightarrow r)\}$ is inconsistent.

The hint is to show that $\displaystyle \neg(q\rightarrow r)\vdash q$ and $\displaystyle \neg(p \rightarrow q)\vdash \neg q$

And of course the hint totally makes sense but I just cannot do it! I feel like I've played around all day with the three axioms and MP and tried using the deduction theorem and deriving things from the empty set and trying to reverse engineer the derivation and UGH! I don't understand how to do these things, is there like a method to figuring it out or is it just reverse engineering and trial-and-error or what?

Any help or hints or anything will be MUCH appreciated!