TITLE: Can a notation for ordered pairs in ZFC set theory be introduced as a conservative extension of ZFC without junk theorems?
QUESTION [6 upvotes]: Currently, a standard way to introduce ordered pairs in ZFC set theory is to define
an ordered pair $(a,b)$ as $\{\{a\},\{a,b\}\}$. From this definition one can deduce the main property of ordered pairs:
$(a,b) = (x,y) \rightarrow a=x \,\&\, b=y$.
But some junk theorems can also be proved, for example, $(a,a) = \{\{a\}\}, \{a\} \in (a,b)$, etc.
see also https://mathoverflow.net/questions/90820/set-theories-without-junk-theorems
https://mathoverflow.net/questions/386756/a-better-way-to-introduce-ordered-pairs-in-zf
For a different approach see the paper "Adding an Abstraction Barrier to ZF Set Theory"
https://link.springer.com/content/pdf/10.1007%2F978-3-030-53518-6_6.pdf
An excerpt from it:
" Such ‘accidental theorems’ do not match how people think about the mathematics and also cause practical difficulties when using set theory in machine-assisted theorem proving."
So a natural question arises: is it possible to introduce ordered pairs in a more safe way, without possibility to get
such strange "theorems".
A rather natural approach is to extend ZFC just by adding a new function symbol (say $\pi$, it is usually omitted)
and a defining axiom such as  $\pi(a,b) = \pi(x,y) \rightarrow a=x \,\&\, b=y$
and prove that such theory extension is conservative (without such proof we can get an undesirable result, in a worse case - an inconsistent theory).
So my question:
Is this extension of ZFC (by adding the function symbol $\pi$ and its defining axiom)  conservative?

REPLY [3 votes]: Whatever we want of ordered pairs is absolutely not only the famous implication. I believe that (a,b) being Goedel-constructible in a,b is a sine qua non in set theoretic practice above 5 or so introductory lectures, and so is (a,b) being easily coded by a real provided a,b are reals, and zillion more such properties. So an abstract non-definable Bourbaki-style pair in the language will not work.