Undecidability and models of ZFC

This is question that probably comes up a lot here, but every time I think I have it straight, I find I don't. So allow me to ask it in the following form. Four elementary facts:

(1)ZFC $\displaystyle \rightarrow$[ if $\displaystyle \kappa $ is an inaccessible cardinal, then ( <$\displaystyle V_{k}$ , $\displaystyle \epsilon$> $\displaystyle \models $ZFC) ]

(2) The existence of an inaccessible cardinal is consistent with ZFC

(3)not [ZFC$\displaystyle \rightarrow $( ZFC is consistent)].

(4) A theory is consistent iff it has a model.

At first glance, (1) and (4) would seem to imply

(*) ZFC$\displaystyle \rightarrow$( ZFC is consistent) which of course is rubbish. One possibility for the problem is that perhaps one cannot assert

(4') ZFC $\displaystyle \rightarrow$ (A theory is consistent iff it has a model),

but I am not sure whether one cannot (since the model concept is formalizable in the language of ZFC), and even if one cannot, I am not sure that that would be the main fissure in the faulty conclusion. I suspect there is something much more basic here that I am missing.