Enderton 2.6.8

Assume that $\displaystyle \sigma$ is true in all infinite models of a theory $\displaystyle T$. Show that there is a finite number $\displaystyle k$ such that $\displaystyle \sigma$ is true in all models $\displaystyle A$ of $\displaystyle T$ for which $\displaystyle |A|$ has $\displaystyle k$ or more elements.

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The sentence for there are at least $\displaystyle k$ things for $\displaystyle k \geq 2$ is

$\displaystyle \lambda_2 \wedge \lambda_3 \wedge \ldots \wedge \lambda_k$, where

$\displaystyle \lambda_k = \exists v_1 \exists v_2 \exists v_3 \ldots \exists v_k( (v_1 \neq v_2 \wedge v_1 \neq v_3 \wedge \ldots \wedge v_1 \neq v_k \wedge v_2 \neq v_3 \wedge \ldots \wedge v_2 \neq v_k \wedge \ldots \wedge v_{k-1} \neq v_k)$

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Any hint to get started?

Thanks.