TITLE: How to define Conway's class of all games rigorously?
QUESTION [7 upvotes]: I'm reading John Conway's On Numbers And Games.
In the course of defining the surreal numbers, Conway defines the term game more or less as follows.

$(\emptyset,\emptyset)$ is a game.
If $L$ and $R$ are sets whose elements are games, then $(L,R)$ is a game.
All games are constructed in this way.

Obviously this is an informal definition, so I'd like to see a formalized version. However, the 'solution' to this problem discussed in the Appendix to Part 0 doesn't make sense to me.
Can someone provide a formal definition of the predicate $\mathrm{Game}$ which returns true iff the input is a game?
Note that the issue here is quite distinct from the issue discussed here: are surreal numbers actually well-defined in ZFC?.

What I've tried so far. It's tempting to proceed as follows. Define that a class $C$ is closed iff whenever $L$ and $R$ are sets with $L,R \subseteq C,$ then $(L,R) \in C.$ Then the class of games is defined as the intersection of all closed classes incorporating $(\emptyset,\emptyset)$.
There's a few issues with this. Firstly, in ZFC, we don't have classes, just predicates. So, how do we rehash the notion of an intersection of all closed classes in terms of predicates? Is it even possible? I'm willing to use NBG or MK if its strictly necessary, but I don't think it is.
Secondly, how do we know there exists a closed class? We don't want to use the cumulative hierarchy to construct such a class, because the whole point of Conway's approach is that we get the ordinals for free as a subset of the surreals.

Edit. In the comments, it was mentioned that $\in$-recursion makes Conway's definition rigorous. Can someone explain precisely how this is done?
For instance, does $\in$-recursion allow us to define 'game' by asserting that $x$ is a game iff there exist sets $L,R$ such that $(L,R)=x$ and the elements of $L$ and $R$ are games.

REPLY [7 votes]: Here's a definition of the notion of "game" in ZF; it's essentially re-doing, for this particular example, the general technique for eliminating recursion in favor of explicit definitions.
I'll begin with an auxiliary definition, that I'll call "evidence"; the intuition is that a set is evidence if it contains enough information to verify that each of its elements is a game.  Formally, a set $E$ is evidence if it is a set of ordered pairs and, for each pair $(L,R)\in E$, both $L$ and $R$ are sets and all their members are also in $E$.  [Technically, I could omit the clause "both $L$ and $R$ are sets," because in ZF everything is a set, but I think the definition is easier to understand with that clause included.]  Note that this is an explicit definition of what it means for a set to be evidence.
Now define a set $G$ to be a game iff it is a member of some $E$ that is evidence.