ordered pair

For any sets $a$ and $b$, the ordered pair $(a,b)$ is the set $\{\{a\},\{a,b\}\}$.

The characterizing property of an ordered pair is:

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

and the above construction of ordered pair, as weird as it seems, is actually the simplest possible formulation which achieves this property.