Let $\Gamma$ be a consistent set of $L$-sentences with infinite cardinality. If it has an infinite model, then there exists a model for $$ \Gamma' =\Gamma \cup \{\lnot c_a = c_b : a \neq b \}, $$ where $c_i$ is a constant symbol.
How would I go about proving this? I'm imagining that I can extend the model for $\Gamma$ to have a larger domain (to include more constant symbols) but I don't know how to show that it is actually a model for $\Gamma'$. Maybe with compactness?
A finite subset of $\Gamma'$ is a finite subset of $\Gamma$, together with the assertions that finitely many $c_1,\dots,c_n$ are pairwise distinct.
Take any infinite model of $\Gamma$ (which exists by assumption), interpret $c_1,\dots,c_n$ as distinct elements, and interpret the rest of the constant symbols arbitrarily (we are free to do this, since the constant symbols are not mentioned in $\Gamma$). This is a model of our finite subset of $\Gamma'$. That's all there is to it.
This is the standard proof of the (upwards) Löwenheim-Skolem theorem.