The question is originally from math stack exchange here.

Basically, what I am asking is if we can define ordered tuples of proper classes in *NBG*. My idea, for finite tuples of proper classes, was to define it as the set $\{0\} \times X_0 \cup \{1\} \times X_1 \times ... \times \{n\} \times X_n$. Here I am not assuming $(X_i)_{i < n+1}$ as an encoded structure since that would assume the conclusion. However, we can define this using recursion. The empty tuple is the empty set(i.e. $()=\emptyset$) and $(X_0, X_1, ..., X_n) = (X_0, ..., x_{n-1}) \cup \{n\} \times X_n$.

Now I would intuitively define $(X_\alpha)_{\alpha < \beta}$ for $ \beta $ an ordinal, by transfinite recursion, by taking $(X_\alpha)_{\alpha < \beta + 1} = (X_\alpha)_{\alpha < \beta} \cup \{\beta\} \times X_{\beta}$ and if $\beta$ is a limit ordinal, then $(X_\alpha)_{\alpha < \beta} = \bigcup_{\alpha < \beta} \left( (X_{\gamma})_{\gamma < \alpha} \right)$. However, there is a problem with the limit ordinals, since I am taking the "infinite union", the sum class, which is basically the union of its members, so it would require that $X_a \in T$, which is impossible since $X_a$ is a proper class.

However, I have also found this answer on stack exchange that uses exactly this method and it I don't know whether what I said above is correct or not.