The Axiom of Extensionality is sometimes know also as the Axiom of Extent or the Axiom of Extension.
∀z(z ∈ X ↔ z ∈ Y) → X = Y
If for all z, z is an element of X if and only if z is an element of Y, then X=Y.
We assume some set Z exists and it contains all the elements z. We call Z the universal set. Every element in X will be in Z and every element in Y will be in Z.
There may be elements in Z which are not in X, and therefore also not in Y.
To show this, we create a simple example:
X={2,4}
Y={2,4}
Z={1,2,3,4,5}
One key thing to notice, the if and only if for (x element of Z) and (y element of Z). This leaves open the door for there to be a z that is not in X and not in Y.
Appendix A
The textbook wrote it as:
∀z(z ∈ x ↔ z ∈ y) → x = y
We replaced x and y with X and Y because we recognize X and Y as being sets given their location to the right of 
Rob Sterling 