# A consequence of the comprehension schema is that the intersection of two sets is a set

I am currently doing a project on set theory and at the minute i am studying ZF axioms. A consequence of the comprehension schema is that the intersection of two sets is a set. Hnece i have come up with the following proporsition.

Proposition: Let $$x$$ and $$y$$ be two sets. Then the intersection of $$x$$ and $$y$$, $$x\cap y$$ is a set.

Proof: Define $$P(z) = z \in Y$$. Then $$x\cap y = \{ z \in x \wedge P(z)\} =\{ z \in x \wedge z \in y \}$$. Hence, by the comprehension schema $$x\cap y$$ is a set.

I would greatly appreciate if someone could let me know if there are any mistakes in the proof.

Yes not only is your proof correct for the intersection of two sets but with a minor adjustment you can prove it for any non empty class of sets (defined by a first order formula). This is intuitively true because the intersection is a subset of any of the sets partaking the intersection. The way you prove it is similar. Given a class (a collection of sets that may or may not be a set itself) defined by the formula $$\varphi(x)$$ then we can choose a set $$a$$ in the class (this is where you need it to be non empty) and we define the intersection to be $$\{x\in a:\forall y (\varphi(y)\rightarrow x\in y)\}$$ The new first order formula which uses comprehension is now $$\psi(x)\equiv \forall y (\varphi(y)\rightarrow x\in y)$$.