I'm working on the following problem which is exercise 3.5.1 in Rothmaler's Model Theory book. Show that theory $T$ is complete iff $\phi \vee \psi \in T$ implies $\phi \in T$ or $\psi \in T$ (keep in mind that in his book, a theory is deductively closed)
My forward direction looks like this: Let $\mathcal{M}$ be a model of T, if $\mathcal{M} \models \phi \vee \psi$ then $\mathcal{M} \models \phi$ or $\mathcal{M} \models \psi$. $\mathcal{M} \models \phi$ means $\phi \in Th(M)$ but because $T$ is complete, $T=Th(M)$ so $\phi \in T$. Similarly for $\psi$. Is this correct?
For a counter example of the forward direction, I was thinking something along the line of: Let $T$ be PA (which is incomplete), and consider non-standard model $\mathcal{M}$ of $T$, then in $\mathcal{M}$, the sentence "$x \neq x$ or there's a biggest elt" is true, but neither $x \neq x$ or "there's a biggest elt" is in PA. Does this make sense?
I'm not sure about the backward direction.