In answer to this post
There is no one fixed way to define an ordered pair in terms of sets. It is also common to define an ordered pair as $(π₯,π¦):=\{π₯,\{π₯,π¦\}\}$. One can prove that $\{x_1, \{x_1,y_1\}\} = \{x_2, \{x_2,y_2\}\} \iff x_1 = x_2 \text{ and }y_1=y_2 \label{1}\tag{$*$}.$
I've tried the proof, and the process is questionable:
take simplest case when
$$
x=y,\, u=v,\, \langle x,y\rangle=\langle u,v\rangle \iff \{π₯,\{π₯\}\}=\{u,\{u\}\},$$
to which $x=u,\{π₯\}=\{u\}$ is a solution. As we can immediately see, also $x=\{u\},u=\{π₯\}$ can be a solution as long as there exist a set such that $x=\{\{x\}\}$, i.e. if $x$ is a set containing a set containing itself.
Now searching the internet for an example of a set containing itself, I've found this Quora Q&A, where it states that in ZFC set theory there isn't a set containing itself (so I think it also means a set containing a set containing itself wouldn't exist). On the other hand,
other non-ZFC set theories allows a set containing itself to exist (so I assume a set containing a set containing itself would also exist under such theory), so under this theory
we cannot prove \eqref{1}.
On the contrary to Kuratowski's definition $\langle x,y\rangle=\{\{x\},\{x,y\}\}$, Wiener's definition
$$
\langle x,y\rangle=\{\{\{x\},\emptyset\},\{\{y\}\}
$$ is a good one because it don't rely on such a particular axiom of set theory. It works in set theories other than ZFC, even works in naive set theory.
Best Answer
As other commenters have pointed out, there are multiple possible viable set-theoretic representations of ordered pairs. But it seems like the OP's specific difficulty is in understanding why the convention of $\{x, \{x, y\}\}$ actually "works", so to speak.
I'd argue it as follows (Iβm assuming a βpureβ set theory, without βurelementsβ): The key is whether, for any instance of $\{x, \{x, y\}\}$, one can unambiguously tell which element is intended as the "first" and which element is the "second". This is indeed possible, because necessarily $x \in \{x, y\}$ but it cannot be true that $\{x, y\} \in x$ (that would violate the Axiom of Foundation). So we can tell which element is the "first", and then trivially we know which is "second" (I've assumed $x$ and $y$ are distinct -- a nearly identical argument pertains if $x = y$)
So $\{x, \{x, y\}\}$ seems like a viable convention for representing ordered pairs, but I think the Kuratowski convention is preferrable in that it's just clearer how it works (as evidenced by OP's perplexity). Also it's not totally obvious to me that $\{x, \{x, y\}\}$ would work for every case in non-well-founded set theories (lacking an Axiom of Foundation)