Past exam of the logic and set theory course of the University of Oxford / 2010 / A / 3 / b / ii

Suppose a structure satisfies every sentence of $\Sigma$. By the soundness theorem for first-order logic, every sentence derivable from $\Sigma$ is true in this structure. In particular it cannot derive both a sentence and its negation, because they cannot both be true under the usual semantics.

Therefore $\boxed{\Sigma \text{ satisfiable}⟹\Sigma \text{ consistent}}$. This direction uses soundness alone, without constructing a model from a proof-theoretic assumption.