This is taken form A. Karr probability book.
a) A d-system is a family of subsets containing Ω and closed under proper difference (if A, B ∈ D and A ⊆ B, then B\A ∈ D) and countable increasing union.
b) A π-system is a family of subsets closed under finite intersection.
A σ-algebra is a d-system and a π-system. In addition, a class that is both a π-system and a d-system is a σ-algebra.
It seems to me that from a) follows b), because "containing Ω and closed under proper difference" imply "closed under complement", and "closed under countable increasing union" imply "closed under finite (and countable) union". Am I missing something? Because, if I'm right, I don’t' understand the need of defining both σ-algebra and d-system.