Let the language of arithmetic $\mathcal L^A$ denote the first-order language with the function symbols $s$ (successor), $+$, and $\times$ and the constant $0$.
Let $TA$ denote the set of sentences $\phi$ of $\mathcal L^A$ such that $\mathbb N \models \phi$. (Here, $\mathbb N$ denotes the standard model of $Q$.)
Finally, we say that a theory $\Sigma$ is maximal-consistent in $\mathcal L^A$ if it's consistent and for any consistent theory $\Theta$ such that $\Sigma \subseteq \Theta$, we have $\Sigma = \Theta$.
It's clear that $TA$ is consistent. However, although it seems obvious as well, I cannot produce a formal proof of the fact that $TA$ is maximal.
Could someone please show me how to do so?