If we take $\sf MK$ and $\sf PMK$, the latter is just a weakening of the former by restricting the formulas in class comprehension to be relativised to sets which is by the way a mono-sorted version of $\sf NBG$. Does this weakening affect what the class $\sf ON$ of all set von Neumann ordinals is?
Is it a theorem of both theories that "$\sf ON \geq icc_0$"?
Where $\sf icc_0$ is the first strongly inaccessible cardinal.
The point is that $\sf MK$ is known to be strictly stronger than $\sf NBG$ in consistency strength, and if the above is correct, then this added strength is not reflected on the least value the class of all ordinals can take in that theory? Does that mean that both theories would have the same proof theoretic ordinal?