Let $A$ be any set which could be considered a tuple — that is, $A = (B, C)$ holds for some sets $B$ and $C$. Then what expression can find $B$ and $C$? I mean that an expression is a set whose existence is guaranteed by the axioms of ZFC. So, what I want is to find $\mathrm{fst}$ and $\mathrm{snd}$ such that $A = \left( \mathrm{fst} A, \mathrm{snd} A \right)$.
Added: We use Kuratowski's definition of ordered pair – i.e., $(a,b):=\{\{a\},\{a,b\}\}$
Best Answer
Given a tuple $T = (x, y) = \{\{x\}, \{x,y\}\}$, observe that $\bigcup T = \{x, y\}$ by the axiom of union. So we would have $$ x = \bigcup \left\{\, a \in \bigcup T : \{a\} \in T \,\right\} $$ using the separation and pairing axioms. We can then use this $x$ to obtain $$ y = \bigcup \left\{\, a \in \bigcup T : (x, a) = T \,\right\} $$ again using the separation and pairing axioms.
There won't be an issue if $x = y$, because then we would have $T = \{\{x\}\}$ and $\bigcup T = \{x\}$.