Wouldn't it be easier to just say that for any set $x$, there exists a set $Y$ such that $x\in Y$? That seems like a simpler statement. Plus for any sets $x, y$, this statement + restricted comprehension gives the sets {$x$}, {$y$}. So union would give us {$x, y$} and we recover the original axiom of pairing.
I have not taken a set theory course.