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Axiom of extensionality

Before giving the axioms that will allow us to construct the integers we give the axiom of set theory that defines what we mean by `$=$'. Without this axiom there would be little point in defining the integers or anything else. The axiom of extensionality tells us that sets are uniquely defined by their members.

$$\forall x \forall y \hspace{.1in} (\forall z \hspace{.1in} z \in x \equiv z \in y) \equiv (x=\index{\protect=}y)$$

$a \equiv b$ means $a$ and $b$ have the same truth value or are equivalent. They are either both true or both false. It is the same as $(a\rightarrow b)\wedge (b \rightarrow a)$. This axiom says a pair of sets $x$ and $y$ are equal if and only if they have exactly the same members.