= Łoś-Tarski preservation theorem
{c}
{title2=$T\equiv T_\forall$}
A <first-order theory> has its class of <first-order models> closed under <substructures of a first-order structure> exactly when it is equivalent to its universal consequences $T_\forall$. If $A\models T_\forall$, then $T\cup\operatorname{Diag}(A)$ is consistent. Otherwise a finite <logical conjunction> $\delta$ of signed <diagram of a structure> sentences yields a universal consequence $\forall\mathbf x\,\neg\delta(\mathbf x)$ false in $A$. The <compactness theorem> gives a <first-order model> of $T$ containing an embedded copy of $A$, and closure under <substructures of a first-order structure> implies $A\models T$. The converse is preservation of <universal sentences>. The <diagram of a structure> must include negated as well as positive atomic sentences so that the resulting map is an embedding, not just a homomorphism.
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