Axiom 3.5

Axiom 3.5. (Axiom of specification). Let \(A\) be a set, and for each \(x\in A\), let \(P(x)\) be a property pertaining to \(x\) (i.e., \(P(x)\) is either a true statement or a false statement). Then there exists a set, called \(\{x\in A:~P(x)\text{ is true}\}\) (or simply \(\{x\in A:~P(x)\})\) for short), whose elements are precisely the elements \(x\) in \(A\) for which \(P(x)\) is true. In other words, for any object \(y\), $$ y\in\{x\in A:~P(x)\text{ is true}\}\iff (y\in A \text{ and } P(y)\text{ is true }). $$