I'll look into this, but first I have a statement and a question.
Statement: Unfortunately, in logic there are many different but equivalent ways to define many concepts. E.g., one can take \/, /\, -> and _|_ (falsehood) as primitive connectives, or take only -> and _|_ and consider the rest of the connectives as abbreviations. To make things worse, there are at least three general ways to define proofs: one (Hilbert's style) deals with formulas and has only inference rule but many axioms, the second (Natural Deduction) also deals with formulas but has many rules and no axioms, and the third (Sequent Calculus) deals with sets of formulas and also has many rules and no axioms. Therefore, if you look at ten different logic textbooks, you will probably find ten different notations, primitive connectives, inference rules, etc., though in the end they all give equivalent logics.
Question: What are the inference rules that were given to you? Is it Natural Deduction (ND)? ND has introduction and elimination rules for each connective, and it has open and closed assumptions. You mentioned \/E (disjunction elimination rule); does it look like this?
The proof that you are looking for depends a lot on the definition of the proof, i.e., inference rules.
A \/ B C C
Edit1: Also, what exactly is Disjunctive Syllogism?