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Axiom of unordered pairs

If we have two sets $x$ and $y$ we can construct a set that contains both $x$ and $y$. The notation for that set is $\{x,y\}$.

This axiom constructs a new set from any two existing sets.

$$\forall x \forall y \hspace{.05in} \exists z \hspace{.05in}\forall w \hspace{.1in}w \in z \equiv ( w = x \vee w = y )$$

This says for every pair of sets $x$ and $y$ there exists a set $w$ that contains $x$ and $y$ and no other members.