I'm taking a introduction to Logic course and came across the following result:
Let $\Theta$ be a theory over a decidable signature $\Sigma$. Assume $\Theta$ has quantifier elimination and $F_0=\emptyset$ (the set of constant symbols). Show $\Theta$ is complete.
This is presented as a direct consequence of the following proposition (which I have been able to prove):
Let $\Theta$ be a theory over a decidable signature $\Sigma$. Assume the theory has quantifier elimination and that for each $\alpha\in A_\Sigma \cap cL_\Sigma$, either $\Theta \models_\Sigma \alpha$ or $\Theta \models_\Sigma \neg {\alpha}$ ($A_\Sigma$ is the set of Atomic formulas and $cL_\Sigma$ the set of closed formulas). Then, $\Theta$ is complete.
I'm missing some basis on FOL and so, some results are really hard to connect. Maybe it's really obvious but I do want to understand why it is a consequence of the second proposition. Thank you in advance!
EDIT
If there are no constant symbols, then the close formulas must be something like $\forall \psi$, which are not atomic formulas and therefore $A_\Sigma \cap cL_\Sigma = \{ \bot\}$. So the previous would hold. Is this the right path?