Proof of Class Comprehension in Two and Three Set Variables, Stated with Ordered Pairs

For (i), apply The Class Comprehension Theorem for Predicative Formulas (i) with $n=2$, $x_{1}=u$ and $x_{2}=v$, to the formula $\varphi(u,v,Y_{1},\dots,Y_{m})$ and the given classes $Y_{1},\dots,Y_{m}$ and assignment of sets to the parameters. It gives a class $Z$ such that, for all sets $u$ and $v$, $\langle u,v\rangle\in Z$ if and only if $\varphi(u,v,Y_{1},\dots,Y_{m})$. By the recursion of The Ordered Pair of Two Sets and Nested Tuples (ii),

$$\langle u,v\rangle=(\langle u\rangle,v)=(u,v),$$

so $Z$ is as required.

For (ii), apply The Class Comprehension Theorem for Predicative Formulas (i) in the same way with $n=3$, $x_{1}=u$, $x_{2}=v$ and $x_{3}=w$. It gives a class $Z$ such that, for all sets $u$, $v$ and $w$, $\langle u,v,w\rangle\in Z$ if and only if $\varphi(u,v,w,Y_{1},\dots,Y_{m})$, and by the same recursion

$$\langle u,v,w\rangle=(\langle u,v\rangle,w)=((u,v),w),$$

so $Z$ is as required.