Let's take the minimal transitive model of $\sf ZFC$ which, I came to know, is some minimal $L_\kappa$ for a countable $\kappa$, that models $\sf ZFC$, and since its minimal so no subset of it can be a transitive model of $\sf ZFC$ and at the same time not isomorphic to it, call this stage $L_{\sf ZFC}$.
What kind of cardinalities $Th(L_{\sf ZFC})$ prove? I know it proves $\sf GCH$ because it's a model of $\sf ZFC + [V=L]$, but what theorems about cardinal existence it proves? Since this theory is complete, then it must decide on for example whether inaccessible cardinals exist or not?
A which large cardinal, $Th(L_{\sf ZFC})$ starts to prove its non-existence?
I tend to think that because it is minimal so it must prove the non-existence of inaccessibles whether strong or weak. Is that correct? And even if so, then what's the exact argument for that? If no, then at which large cardinal it starts proving its absence?