At least, here's the argument that opened the question for me:
The anticlass-theory principle: there are no discrete proper classes. There are intensional elementhood parameters such that if some set X satisfies those parameters in all such cases, then it satisfies all other such intensional parameters too. For example, if |ORD| = CARD then the well-ordering lemma follows about as trivially as could be desired, and hence as much as is not desirable.j:V→V? But according to the anticlass principle, if X contains all ordinals, then it contains all cardinals, as well as as surreals and so on and on. There is no class that contains all and only ordinals or cardinals or surreals, etc. (The universal exclusifier is compromised to avoid the paradoxes of naive comprehension.)
With respect to talk of the surreal numbers, there are negative shadows of all positive transfinite ordinals. (Then there are things like the roots of negative transfinite surreals, etc.) So we can talk about a set of all negative numbers whatsoever. If the universal positive set is V, let's call this ultimate shadow by the name of V-minus. (This is, incidentally, a cofounded set, since it satisfies its own parameters for inclusion as an element.)
However, if there are "as many" negative transfinite surreals (surdinals?) as positive transfinite ordinals, then either |SURD-minus| = |ORD-plus| (on the class-theoretic scheme of things) or V itself contains all negative numbers, and is the only set that happens to contain all negative numbers.
Ergo, V equals V-minus, or V = -V. QED
Counterargument: 0 itself already admits of +0 = -0. V is (Fregewise) the equivalent of "anti-zero," but shares some peculiar traits of 0. Actually, historically, it has often been given that all true formulae are vacuously true of the nonexistent elements of the empty set, and all true formulae are vacuously true within the unrestricted transet also. So perhaps there isn't anything more contrary to the law of identity about V = -V than there is about 0 = -0.
Still, Frege's introduction of "anti-zero" was to invoke a term that covered all instances of x = x. Here, we are saying that this action means trying to cover a universal instance of x = -x too, though, which might be seen as contrary to the purpose of Frege's introduction.
EDIT: for clarity, I have to distinguish writing the string {x = -x} from writing {x = not-x} and {x = anti-x}. The anticlass argument seems to end up with {V = anti-V}, except if V is anti-zero, then wouldn't anti-{anti-zero} be zero? So we'd be saying V = 0, it looks like, which is still problematic.
j:V→V?The derivation is kind of... cute? Hamkins covers a series of propositions equivalent to the axiom of global choice. One is that |ORD| = V, another is that for all proper classes Ca, |Cn| = |Cm|. But so if it seems possible to negate these propositions, we can get |ORD| < CARD or maybe rather |ORD| ≹ CARD. But if it is possible for |ORD| to be < CARD, then it is possible that |ORD| > CARD, too, which seems absurd. However, suppose that neither ORD nor CARD "exist" in the first place. Alternatively, treat with expressions like ORD and CARD like you would divisions-by-zero in normal arithmetic: invalid, or beyond-undecidable, or whatever along that line. At any rate, if there are no ORD or CARD in play, then, "Does |ORD| = CARD?" is not a (locally) well-grounded question. Then even though there is a "virtual" sense in which there are absolutely infinitely many ordinals and cardinals (and surreals!) and whatever else besides, or it is "virtually" as if all proper classes are the "same size," this is not so in a way that authorizes the derivation-by-translation of the global choice axiom. I'm not sure why I'm so exercised to make this point, although it does play into considerations of "separation-minus" or "inexcisable" sets (sets such that they have separation-resistant elements, from whom certain kinds of subsets are not possible to excise); but the other point is, still, that if there are any proper classes, or unrestricted transets, or whatever else as such, these things will contain everything in every way as just one class/transet/etc. Even the unrestricted multiset is the same thing: since the absolute transet satisfies every set-categorized parameter, it satisfies the concepts of multisets, fuzzy sets, and so on and on as well. But this definitely requires handling the universal exclusifier, "all and only," carefully.