1. T ⊃ O
2. R v ~O
3. ~R              /~T

4. ~O            2,3 Disjunctive Syllogism | You Must Learn: [(p v q) & ~p] ⊃ q
5. ~T            1,5 Modus tollens         | You Must Learn: [(p ⊃ q) & ~q] ⊃ p

It's common sense that if you know one of two things is true, then if one of them
is not true, then the other one must be true.  That's disjunctive syllogism!

It's also common sense that if a first thing implies a second thing, and the
second thing is not true, then the first thing must not have been true either.
That's modus tollens.

All logic is plain old common sense.  You just have to learn what it says.

Edwin