axiom of pairing

For any $a$ and $b$ there exists a set $\{a,b\}$ that contains exactly $a$ and $b$.

The Axiom of Pairing is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads:

$$\forall a\forall b\exists c\forall x(x\in c\leftrightarrow x=a\vee x=b).$$

Using the Axiom of Extensionality, we see that the set $c$ is unique, so it makes sense to define the pair

$$\{a,b\}=\text{ the unique }c\text{ such that }\forall x(x\in c\leftrightarrow x=a\vee x=b).$$

Using the Axiom of Pairing, we may define, for any set $a$, the singleton

$$\{a\}=\{a,a\}.$$

We may also define, for any set $a$ and $b$, the ordered pair

$$(a,b)=\{\{a\},\{a,b\}\}.$$

Note that this definition satisfies the condition

$$(a,b)=(c,d)\text{ iff }a=c\text{ and }b=d.$$

We may define the ordered $n$-tuple recursively

$$(a_1,\mathrm{\dots },a_n)=((a_1,\mathrm{\dots },a_{n-1}),a_n).$$