Logic: $\text{Mod } \Sigma$, the class of all models of $\Sigma$ and $\text{Th Mod }\Sigma$, how do these relate?

Logic: $\text{Mod } \Sigma$, the class of all models of $\Sigma$ and
$\text{Th Mod }\Sigma$, how do these relate?

While reviewing a question I had asked earlier here: Proving and
understanding the Fixed point lemma (Diagonal Lemma) in Logic - used in
proof of Godel's incompleteness theorem
I have the following:
"$\text{Mod } \Sigma$ is the class of all models of $ \Sigma$. $\text{Th
Mod } \Sigma$ is the set of all sentences which are true in all models of
$\Sigma$. This however is just the set of all sentences logically implied
by $\Sigma$. We call this set the set of consequences of $\Sigma$ or
$\text{Cn }\Sigma$. Thus we have that $\text{Cn }\Sigma = \{\sigma \mid
\Sigma \models \sigma \} = \text{Th Mod } \Sigma$."
Letting $\Sigma$ denote a set of axioms. Since $\text{Th Mod } \Sigma$ is
the set of all sentences which are true in all models of $\Sigma$ then
this seems to imply that there are sentances which may not be true in all
$\text{Th Mod } \Sigma$. So in other words $\text{Th Mod } \Sigma$ is
derivable from $\Sigma$ i.e. we can build some sentence in $\text{Th Mod }
\Sigma$ from axioms in $\Sigma$. I however am trying to understand better:
$\text{Th Mod } \Sigma$ is implied from $\Sigma$. This would mean that
$\Sigma \Rightarrow \text{Th Mod } \Sigma$. Now this is always true (is a
Tautology) as the only case which would be false would be that something
derived from a set of axioms (or just a set of axioms?) in $\Sigma$
implying $\text{Th Mod } \Sigma$ would be false as all statements in
$\text{Th Mod } \Sigma$ are true by definition. I am thinking this
explains their relationship, but could use some confirmation and maybe
some more insight.
Thanks,
Brian