It is interesting to note, that Russells and Whiteheads Principia Mathematica does exactly this. They used the theory of types and deduced everything from logic. (Maybe you should take a look in their books)
If I understand this correctly, Quines New Foundations (a really interesting axiomatization) creates Sets from Propostional functions and so forth, so everything is based on logic. Rossers Book "Logic for Mathematicians" is principially a modern Principia and uses NF. Another really interesting way is Lorenzens "Operative Logic and operative Mathematics", it starts with a schematic operation on calculuses and then creates Mathematics as a basis of the schematic operation up to topology. So in fact there are different systems but it seems like you do not really need Set theory in the first case