I think it is important to consider. The axiom of extensionality is provided as a means of defining equality between sets. Roughly stated we say *"A set X is equal to set Y if and only if they both have exactly the same elements."* A more precise statement, taking into account that we'd like to quantify over domains that we can define, is this:

1) $X=Y\leftrightarrow \ \forall x \in X\ (x\in Y) \ \wedge \ \forall y \in Y\ (y\in X).$

This is just defining an equality relation for sets, and reflexivity comes from the definition rather than having to be be put forth as an axiom. For example taking *X=X*,

$\forall x \in X\ (x\in X) \ \wedge \ \forall x \in X\ (x\in X),$

is pretty true.

We can say that reflexivity is a theorem here. It's one of the properties of an *Equivalence Relation*, which must obey reflexivity, symmetry, and transitivity. Our 'equals' relation on sets happily obeys these rules. I like to consider equality as a possible relation we can define about objects, with some properties that we want it to hold. If we were coming at things from the perspective of logic we might make these properties axioms and prove the existence of such relations as theorems. I'm not completely sure on this. But from the perspective of axiomatized set theory, the extensionality axiom implies the rest of the equality properties very concretely.