A Question about the Definition of Ordered Pair [duplicate]

I am self-studying set theory, following the book "The Joy of Sets". It says that an ordered pair $(a,b)$ is defined as the set $$\left\{\left\{a\right\},\left\{a,b\right\}\right\}.$$

Is this definition unique? Why cannot it be defined as $\left\{a,\left\{a,b\right\}\right\}$ or $\left\{a,\left\{b\right\}\right\}$?

Solution 1:

Any definition of $(a,b)$ is fine as long as it satisfies

$$\forall a,b,c,d: (a,b)=(c,d)\iff (a = c) \land (b=d)$$

This definition, due to Kuratowski, is just the traditional one that we can define from just the pairing axiom (so with minimal axioms needed).

Others will do too, see Wikipedia for more.