The definition of the ordered pair is cleverly chosen. If we want to say " is on the first position in " in set-theoretical language, we just say " is an element of all elements of ". If we want to say " is on the second position in " , we say "there exists an element of that contains AND it is not true that there are two distinct elements of both containing ".
Moreover, the characteristic property of oredered pairs, " if and only if and " remains true if we use this definition: " if and only if and " (try to verify it).
This definition comes from Kuratowski and is not the only possible one, but people like it for its properties and have chosen it.