Claim: Suppose T is model complete and has a model embeddable in every model of T. Show T is complete.
This is in Sacks' book Saturated Model theory, problem 8.4.
Is the following proof correct?
Proof: Suppose T is model complete & $\forall$M|= T & $\exists$f, N s.t. f: N|-->X $\subset$M, with f an elementary monomorphism. Thus by assumption, $\forall$ $\phi$$\in$T ($\phi$ has no free variables)N|=$\phi$ iff X|= f($\phi$). Thus M|= f($\phi$); since choice of X, $\phi$ was arbitrary, we have X|= f($\phi$) iff M|= f($\phi$), thus M is elementarily equivalent to X. So $\exists$g: M|-->X & is an e.m. By problem 7.7. T is complete iff $\exists$f, g which are e.m. between models of T having the same range, (and f,g above have the same X), so T is complete. END OF PROOF.
Or maybe I'm supposed to use Robinson diagrams?
Thanks!