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