Condensed representations have been studied extensively for 15 years. In particular, the maximal patterns of the equivalence classes have received much attention with very general proposals. In contrast, the minimal patterns remained in the shadows in particular because they are too numerous and they are difficult to extract. In this paper, we present a generic framework for exact and approximate minimal patterns mining by introducing the concept of minimizable set system. This framework based on set systems addresses various languages such as itemsets or strings, and at the same time, different metrics such as frequency. For instance, the free, δ-free and the essential patterns are naturally handled by our approach, just as the minimal strings. Then, for any minimizable set system, we introduce a fast minimality checking method that is easy to incorporate in a depth-first search algorithm for mining the δ-minimal patterns. We demonstrate that it is polynomial-delay and polynomial-space. Experiments on traditional benchmarks complete our study by showing that our approach is competitive with the best proposals.