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.