Set-valued set theory. II

3* Development of the Elementary Theory First note that the following concepts were defined in section 2 in the process of developing the axioms: subset, null set, strong pair, weak pair, strong unit set, weak unit set, ordered pair, function, domain, range, into, onto, standard union, union, strong power set, power set, strong Cartesian product, and Cartesian product. In addition, the two concepts degree and standard, which have no classical counterparts, were introduced. In this section, we first need to define the other concepts common in elementary classical set theory and then to verify that the sets given by the various definitions have the usual properties, including existence. (The reader is reminded that 3\y means that there exists a unique y: there is such a y and any two are equal.)