I'm trying to comprehend the solution to this problem:
Let $\lambda_{n}$ be defined: $$ \exists v_{1} \ldots v_{n} \bigwedge_{i \neq j, 1 \leq i, j \leq n} \neg\left(v_{i} \doteq v_{j}\right) $$ A theory is a set of $\mathcal{L}$ -sentences that is closed under deduction. A theory $T$ is Henkin if for every $\mathcal{L}$ -sentence of the form $\exists v \phi(v)$ there is a constant symbol $c$ in the language such that $\exists v \phi(v) \rightarrow \phi(c) \in T$. Suppose that the language consists of only two constant symbols $c_{1}$ and $c_{2}$. Prove that $\left\{\theta:\left\{\lambda_{2}, \neg\left(c_{1} \doteq c_{2}\right)\right\} \vdash \theta\right\}$ is Henkin.
Official Solution: Clearly $\left\{\theta:\left\{\lambda_{2}, \neg\left(c_{1} \doteq c_{2}\right)\right\} \vdash \theta\right\}$ is a theory. Let $\mathfrak{A}$ be a set of size 2 then $\mathfrak{A} \models\left\{\lambda_{2}, \neg\left(c_{1} \doteq c_{2}\right)\right\}$ if we interpret $c_{1}$ and $c_{2}$ as two different elements of $A$. So it is consistent. Now any formula of the form $\exists v \phi(v)$ is satisfied by $\mathfrak{A}$ and hence $\phi\left(c_{1}\right)$ or $\phi\left(c_{2}\right)$. Therefore Henkin.
Okay, if a set of $\mathcal{L}$-sentences has a model, it is consistent. I get that. But why does any formula of the form $\exists v \phi(v)$ be satisfied by $\mathfrak{A}$? Furthermore, how can I conclude that $\exists v \phi(v) \rightarrow \phi(c) \in T$ for all $\mathcal{L}$-sentences $\exists v \phi(v)$?
The model $\mathfrak A$ has two objects. But nothing more than this is known about it.
But also the language $\mathcal L$ has no non-logical symbols except for the constant $c_1$ and $c_2$.
Thus, the only atomic formulas that we can form arelike: $x=y, x=c_1, y=c_2, c_1=c_2$ and so on.
Every formula of the language with one free variable $\phi(v)$ expresses a property, but the only "properties" that we can express with the language are boolean combinations of the atomic formulas above, that basically amounts to saying that something is equal to $c_1$ of to $c_2$ or it is different from...