So there is a poster in my math classroom with the axioms of the ZFC laid out but the axiomatization does not include separation, but rather has the axiom of the null set. With separation, I understand how to show that the intersection of two sets exist. However, I don't know how to do it without separation. The poster states the axiom of replacement as: $$\forall X \forall x [x \in X \ \wedge \exists!y (P(x,y))] \rightarrow \exists Y \forall y [y \in Y \leftrightarrow \exists x[x \in X \ \wedge P(x,y)]$$
where $P$ is a predicate containing $x$ and $y$ but not $Y$. So I want to prove that $A \cap B$ exists, ideally Fitch style. I was thinking perhaps that to begin with the power sets of $A$ and $B$ to ensure a non-empty intersection. Then after finding the intersection of the power sets, I can apply the Axiom of Union to construct $A \cap B$. I just can't think of what $P$ should be.
