From this we get singletons (when a = b) and unordered pairs. For any set A, there exists a set whose members are exactly the members of members of A. [ \forall A \exists U \forall x [x \in U \leftrightarrow \exists y (x \in y \land y \in A)] ]
This ensures that a set is determined solely by its elements. There exists a set with no members. [ \exists x \forall y (y \notin x) ]
: The union of two sets is a set.
From this we get singletons (when a = b) and unordered pairs. For any set A, there exists a set whose members are exactly the members of members of A. [ \forall A \exists U \forall x [x \in U \leftrightarrow \exists y (x \in y \land y \in A)] ]
This ensures that a set is determined solely by its elements. There exists a set with no members. [ \exists x \forall y (y \notin x) ] suppes axiomatic set theory pdf
: The union of two sets is a set.
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