Although not formally necessary, besides the symbol \(\in\) one normally uses for convenience other auxiliary defined symbols. For example, \(A\subseteq B\) expresses that \(A\) is a subset of \(B\), i.e., every member of \(A\) is a member of \(B\). Other symbols are used to denote sets obtained by performing basic operations, such as \(A\cup B\), which denotes the union of \(A\) and \(B\), i.e., the set whose elements are those of \(A\) and \(B\); or \(A\cap B\), which denotes the intersection of \(A\) and \(B\), i.e., the set whose elements are those common to \(A\) and \(B\). The ordered pair \((A,B)\) is defined as the set \(\{ \{ A\},\{ A,B\}\}\). Thus, two ordered pairs \((A,B)\) and \((C,D)\) are equal if and only if \(A=C\) and \(B=D\). And the Cartesian product \(A\times B\) is defined as the set of all ordered pairs \((C,D)\) such that \(C\in A\) and \(D\in B\). Given any formula \(\varphi(x,y_1,\ldots ,y_n)\), and sets \(A,B_1,\ldots ,B_n\), one can form the set of all those elements of \(A\) that satisfy the formula \(\varphi(x,B_1,\ldots ,B_n)\). This set is denoted by \(\{ a\in A: \varphi(a,B_1,\ldots ,B_n)\}\). In ZF one can easily prove that all these sets exist. See the  Supplement on Basic Set Theory for further discussion.