In this article, we present an original use of Redescription Mining (RM) for discovering definitions of classes and incompatibility (disjointness) axioms between classes of individuals in the web of data. RM is aimed at mining alternate descriptions from two datasets related to the same set of individuals. We reuse this process for providing definitions in terms of necessary and sufficient conditions to categories in DBpedia. Firstly, we recall the basics of redescription mining and make precise the principles of our definitional process. Then we detail experiments carried out on datasets extracted from DBpedia. Based on the output of the experiments, we discuss the strengths and the possible extensions of our approach.