TITLE: If a theory has a model, does a theory have to be consistent?
QUESTION [5 upvotes]: This may be a too obvious question, but then in completeness theorem, direction is only in one direction: if a theory is consistent, then it has a model. Can we make it stronger and say that a theory has a model if and only if a theory is consistent?

REPLY [9 votes]: Yes; the other half you're asking about is the soundness theorem, which states that if $T$ proves $\varphi$ then $\varphi$ is true in every model of $T$. In particular, if $T$ proves $\perp$ then $T$ has no model, and so by the contrapositive any satisfiable theory is consistent.
The proof of the soundness theorem is much simpler than that of the completeness theorem: we just show that each of the clauses of our proof system match up appropriately with the definition of satisfaction.