I'm trying to show that a theory $T$ is complete if and only if $Th(M) = T^\vdash$ for some model $M$ where $Th(M)$ denotes the set of all sentences that are true in $M$. What I have so far:
$\implies$: Let $T$ be complete. Then for every sentence $\varphi$ of the language $L$, either $T \vdash \varphi$ or $T \vdash \lnot \varphi$. If $\varphi \in T^\vdash$ then by soundness, $\varphi \in Th(M)$. If $\varphi \in Th(M)$ then since $T$ is complete, $\varphi \in T^\vdash$.
$\Longleftarrow$: Let $M$ be some model of $T$ such that $T^\vdash = Th(M)$. Let $\varphi$ be any formula of $L$. We want to show that either $T \vdash \varphi$ or $T \vdash \lnot \varphi$.
Here's where I'm stuck. How can I finish the other direction of the proof? Thanks for your help.