I have trouble working out a proof in the second part of

> Jean-Pierre Ressayre and Alex Wilkie.  <b>Mod&egrave;les non standard en arithm&eacute;tique et th&eacute;orie des ensembles</b>.  Publications Math&eacute;matiques de l'Universit&eacute; Paris&nbsp;VII, 1987.

On page 140, Ressayre writes:

> <b>4.6&nbsp;Th&eacute;or&egrave;me</b> [&hellip;] &ndash;&nbsp;b&nbsp;&ndash;&ensp;En revanche, l'&eacute;nonc&eacute; plus faible &ldquo;$\forall\gamma\ \exists(\mathrm L_{\alpha_i})_{i\leqslant\gamma}$&nbsp;cha&icirc;ne faiblement $\Pi_n$-&eacute;l&eacute;mentaire&rdquo; n'est pas $\omega$-cons&eacute;quence de $\Sigma_{n+1}$-$(\text{collection}+\text{fondation})$.

> <b>Preuve de&nbsp;(b)</b>: on applique 4.5 dans un $\omega$-mod&egrave;le non standard&nbsp;$M$.

This refers back to page 139 on which he writes:

> <b>4.5 Corollaire</b> &ndash; Pour tout mod&egrave;le d&eacute;nombrable non standard&nbsp;$M$ de $\Pi_n$-collection, et tout ordinal non standard&nbsp;$\rho$ de&nbsp;$M$, il existe $I\subset^{\rm e}\mathrm{On}^M$ tel que $\rho\in I$ et $\mathrm L_I^M\models\Pi_n\text{-collection}+{}$&ldquo;il n'existe pas de cha&icirc;ne $\Pi_n$-&eacute;l&eacute;mentaire $(\mathrm L_{\alpha_i})_{i\leqslant\rho+\rho}$&rdquo;

Perhaps I should explain some of the terms used here.

 - $I\subset^{\mathrm e}\mathrm{On}^M$ means $I$&nbsp;is a proper initial segment of&nbsp;$\mathrm{On}^M$ and $\mathrm{On}^M\setminus I$ has no minimum element.
 - If $I\subset^{\mathrm e}\mathrm{On}^M$, then $\mathrm L_I^M=\bigcup_{\alpha\in I}\mathrm L_\alpha^M$.
 - _$\Pi_n$-collection_ denotes the scheme consisting of (extensionality, pair, union, foundation, $\Delta_0$-separation, and) all sentences of the form
     $$\forall a,\bar c\bigl(\forall x\in a\ \exists y\ \theta(x,y,\bar c)\rightarrow
       \exists b\ \forall x\in a\ \exists y\in b\ \theta(x,y,\bar c)\bigr)$$
   where $\theta\in\Pi_n$.
 - _$\Gamma$-foundation_ is the scheme saying &ldquo;every nonempty parametrically $\Gamma$-definable class/set has an $\in$-minimal element&rdquo;.
 - A chain $(\mathrm L_{\alpha_i})_{i\leqslant\gamma}$ is _(weakly) $\Pi_n$-elementary_ if $\mathrm L_{\alpha_i}\prec_{\Pi_n}\mathrm L_{\alpha_\gamma}$ for all $i<\gamma$.

Corollary&nbsp;4.5 visually gives a model of $\Pi_n$-collection, or equivalently, $\Sigma_{n+1}$-collection.  It is also easy to verify that this model satisfies $(\Sigma_n\cup\Pi_n)$-foundation by elementarity.  It is, however, not clear to me how to get a model of $\Sigma_{n+1}$-foundation out of&nbsp;4.5, and I see no reason why $(\Sigma_n\cup\Pi_n)$-foundation should imply $\Sigma_{n+1}$-foundation.  From how Ressarye writes about it, the proof is apparently straightforward (if not immediate).

Does anyone have any idea of how an argument showing $\Sigma_{n+1}$-collection in Theorem&nbsp;4.6(b) above can go?