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:

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

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

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

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

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

We may define the ordered $n$ -tuple recursively

$(a_{1},\ldots,a_{n})=((a_{1},\ldots,a_{n-1}),a_{n}).$