Axioms of Class Existence (Gödel's Axioms B1-B7)

Let $X$ and $Y$ be classes. In what follows $u,v,w$ range over sets, $(u,v)$ is the ordered pair and $\langle u,v,w\rangle=((u,v),w)$ the tuple.

¶ (B1) There is a class $E$ such that, for all sets $u$ and $v$, $(u,v)\in E$ if and only if $u\in v$.

¶ (B2) There is a class $Z$ such that, for every set $u$, $u\in Z$ if and only if $u\in X$ and $u\in Y$.

¶ (B3) There is a class $Z$ such that, for every set $u$, $u\in Z$ if and only if $u\notin X$.

¶ (B4) There is a class $Z$ such that, for every set $u$, $u\in Z$ if and only if there is a set $v$ with $(u,v)\in X$.

¶ (B5) There is a class $Z$ such that, for all sets $u$ and $v$, $(u,v)\in Z$ if and only if $u\in X$.

¶ (B6) There is a class $Z$ such that, for all sets $u$, $v$ and $w$, $\langle u,v,w\rangle\in Z$ if and only if $\langle v,w,u\rangle\in X$.

¶ (B7) There is a class $Z$ such that, for all sets $u$, $v$ and $w$, $\langle u,v,w\rangle\in Z$ if and only if $\langle u,w,v\rangle\in X$.