Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on φsymbol, ε, =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of φsymbol, ε, =, …, η, where … is the tuple operator and η is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of φsymbol, ε, =, …, η, obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving η is allowed.