Ordered tuples of proper classes From time to time I encounter notation like this:

A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ...

The confusing part is that a proper class is used as a component of an ordered tuple (that eventually supposed to have some encoding via nested sets à la Kuratowski pair). Thus, a proper class becomes an element of a set, that is formally impossible.
So, I suppose, this is some sort of abuse of notation. What would be a formally correct way to express such things (in $\sf ZFC$, for example)?

Let $\langle a,b \rangle=\{\{a\},\{a,b\}\}$ denote a Kuratowski pair. Suppose we have $n$ classes (some of them can be proper classes):$$C_i=\{x\mid\phi_i(x)\},\ 1\le i\le n.$$
Our goal is to find a class that could represent an ordered tuple of these. We can use the following class for this purpose:
$$\langle\!\langle C_1,\,...,\,C_n\rangle\!\rangle=\{\langle0,n\rangle\}\cup\bigcup_{1\le i\le n}\{\langle i,x\rangle\mid x\in C_i\}.$$
It is a proper class iff any of its components $C_i$ are proper classes, but all its elements are sets, tagged so that we are able to unambiguously reconstruct the length of the tuple and each of its components:
$$C_i=\{x\mid \langle j,x \rangle\in\langle\!\langle C_1,\,...,\,C_n\rangle\!\rangle\land i=j\}$$
