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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.

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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 $\psi(x)\equiv \forall y (\varphi(y)\rightarrow x\in y)$.

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