TITLE: Morse-Kelley set theory consistency strength
QUESTION [20 upvotes]: I've come across several references to MK (Morse-Kelley set theory), which includes the idea of a proper class, a limitation of size, includes the axiom schema of comprehension across class variables (so for any $\phi(x,\overline y)$ with $x$ restricted to sets, there a class $X=(x : \phi(x,\overline y))$).
I have seen various statements about MK and how it proves the consistency of various things, including $Con(ZF)$, $Con(ZFC)$, $Con(NBG)$, and in fact, for any $T\subset MK$ finitely axiomatized, it proves $Con(T)$.
However, and quite frustratingly, I don't see any references to back up these claims, except occasionally links to other places where the claim was made, but not proven (or even proof-sketched).  I would really appreciate a reference where I can see a proof of these claims, or (if it's easier) a quick sketch of why it should be true.
It's not obvious to me at all why quantifying across proper classes should allow this sort of thing, since all relevant sets (sets of proofs, or sets of statements, or whatever) should be contained in some subset of $\omega$, so should be able to be constructed in $ZF$.

REPLY [13 votes]: Concerning the fact that Kelley-Morse set theory proves Con(ZFC) and much more, I wrote a post on my blog explaining one way to do it.

Kelley-Morse implies  Con(ZFC) and much more

The basic outline of the proof is to show that KM proves the existence of a class truth predicate Tr for first-order truth (and this is what Andreas also suggests in his answer), and then to prove that this truth predicate decrees that each axiom of ZFC is true. It follows by reflection that there will be rank initial segments $V_\theta$ that has the same satisfaction relation, and so these are transitive models of ZFC, and indeed KM proves that the universe is the union of a closed unbounded elementary tower $$V_{\theta_0}\prec V_{\theta_1}\prec\cdots\prec V_\lambda\prec\cdots\prec V.$$
Alternatively, a more syntactic argument proceeds by observing that the truth predicate is closed under deduction, complete and contains no explicit contradictions, and so it provides a complete consistent extension of ZFC.
Follow the link for the details.