Upward Löwenheim–Skolem, including elementary extensions

Expand $L$ by one diagram name for each element of $M$ and by distinct fresh symbols $d_{\alpha }$ for $\alpha <\kappa$. Its size is at most $\kappa$ by the two given bounds and F4. Let $S$ be the elementary diagram together with all $d_{\alpha }\mathrm{\ne }{d}_{\beta }$ for $\alpha \mathrm{\ne }\beta <\kappa$.

A finite subset of $S$ mentions finitely many of the $d$ symbols. The infinitude of $M$ permits interpreting them distinctly: after choosing fewer than their finite number of distinct elements, another exists because $M$ is not that finite set. Interpret diagram names by their named elements, and all other $d$ symbols by one fixed element of $M$. This expansion satisfies the finite subset, because all its diagram sentences hold and all its listed inequalities hold. No requirement says the new $d$ values avoid the old named elements.

F2 under A1 supplies $N\models S$ of size at most $\kappa$. The map $\alpha \mapsto d_{\alpha }^N$ is injective by the inequalities, so $\mathrm{\mid }N\mathrm{\mid }\ge \kappa$ and therefore $\mathrm{\mid }N\mathrm{\mid }=\kappa$. F1 gives an elementary embedding of $M$ into the $L$-reduct of $N$ and its transported copy as a literal elementary extension. The transport is a bijection, so size remains $\kappa$.

For an infinite model $M$ of a countable-language theory and any infinite cardinal $\lambda$, if $\lambda \ge \mathrm{\mid }M\mathrm{\mid }$ apply step 3.1 with $\kappa =\lambda$; the language bound holds because $\mathrm{\mid }L\mathrm{\mid }\le {\mathrm{\aleph }}_0\le \lambda$. If $\lambda \le \mathrm{\mid }M\mathrm{\mid }$, apply F3 with empty parameter set. An elementary extension or substructure satisfies the same sentences as $M$, hence is a model of the theory. Equality of cardinals permits $M$ itself. This proves every infinite target size.