Non-dense sets of subvarieties in a power of an elliptic curve

Let V be an algebraic variety embedded in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of V which are of bounded height and which satisfy certain algebraic conditions are a non-dense subset of V. This result has implications in the context of the Pink-Zilber Conjecture and Mordel-Lang plus Bogomolov Theorem.