The axioms:
My work:
Extension:$$\forall x\forall y,\,(x=y)\iff(\forall z,\,(z\in x\iff z\in y))$$
Empty Set:$$\exists x|\forall y,\,\neg(y\in x)$$
Pair Set:$$\forall c\forall d,\,\exists e|(c\in e)\wedge(d\in e)\wedge[\forall f,\,\neg((f=c)\vee(f=d))\implies\neg(f\in e)]$$
If you consider any...