How to prove the idempotence of consequence operator?

Question

Given language $\mathcal{L}$, $A$ is a set of formulas in $\mathcal{L}$, $\alpha$ is a formula in $\mathcal{L}$, $Th(A)\overset{\text{def}}{=}\{\alpha \in \mathcal{L} \mid A \models \alpha\}$. Here cs491 the paper states Th(A) has a property that $Th(Th(A))=Th(A)$. However, I don't get the proof of this property. I have tried to prove this equality by mutual subset but failed. Any ideas on this are appreciated.

Answer 1

Hint

$\alpha \in \text {Th}(\text {Th}(A))$ means that $\text {Th}(A) \vDash \alpha$.

This means that every model of $\text {Th}(A)$ is a model of $\alpha$.

By definition of $\text {Th}(A)$, every model of $A$ is a model of $\text {Th}(A)$.

Thus, every model of $A$ is a model of $\alpha$, i.e. $A \vDash \alpha$.