Below is a simple proof in propositional logic to show you that the lemma pV¬p can be derived from no givens.

This is a very good example to show you that it is often better to prove backwards . Working backwards as shown, the PC rule is used as there is no givens and this would provide you with a given in the form of an assumption. This is a powerful technique to use in case of no givens. This generates a box with ¬(pV¬p) as an assumption and bottom as the goal. Then Not Elimination rule backwards is applied and click on the assumption. We could have also chosen to use Bottom Introduction rule, the reason we chose to use the Not ELimination rule is that the assumption is a not formula and Not elimination is the best rule to choose under this circumstance.This then gives you pV¬p. You might wonder that we have came back to the start and what have we accomplished by the above steps. The answer is that we are an assumption better off than we were which is good as we can now use this assumption to prove our goal.

Then the OR Introduction rule is used backwards as we have an OR formula as the goal and the options we have are PC or OR Introduction, PC rule cannot be used as it does not provide you with anything new, hence we use OR Introduction. When asked to choose a new goal, ¬p is chosen as shown below (choosing p would have taken us longer to prove the goal). Next we chose to use Not Introduction rule rather than PC rule as using the PC rule would have taken a few more steps to prove. So Not Introduction is used backwards, this generates a new box with p as an assumption and bottom as the new goal.

Then the Not Elimination rule is used backwards as explained before and click on the first assumption ¬(pV¬p). This gives you the new goal pV¬p. Then OR Introduction is used backwards and p is chosen as the new goal, so that it can be ticked by using the Tick rule backwards on the assumption as shown below.

Note: When a rule is said to be used backwards it means click both the goal line and the rule.