Axiom:Axiom of Specification

For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.

$$\forall z: \forall P \left({y}\right): \exists x: \forall y: \left({y \in x \iff \left({y \in z \land P \left({y}\right)}\right)}\right)$$

It is sometimes called:

The Axiom of Specification;

The Axiom of Comprehension;

The Axiom of Selection;

The Axiom of Separation (the latter can be confused with the separation axioms of Hausdorff, which arise in topology, so it's not often used).

This means that if you have a set, you can create a set that contains some of the elements of that set, where those elements are specified by stipulating that they satisfy some (arbitrary) condition.

In the above, $$P \left({y}\right)$$ is any function of propositional logic which returns either true or false depending on what $$y$$ is.

Alternatively, this axiom can be specified as follows:

"If $$\phi$$ is a property (with parameter $$p$$), then for any $$X$$ and $$p$$ there exists a set $$Y = \left( {u \in X: \phi \left({u, p}\right)}\right)$$ that contains all those $$u \in X$$ that have the property $$\phi$$."

$$\forall X: \forall p: \exists Y: \forall u: \left({u \in Y \iff \left({u \in X \land \phi \left({u, p}\right)}\right)}\right)$$