An accessibility theorem for infinite graph minors

We prove the following recent conjecture of Halin. Let $\Gamma_0$ be the class of all graphs, and for every ordinal $\mu>0$ let $\Gamma_\mu$ be the class of all graphs containing infinitely many disjoint connected graphs from~$\Gamma_\lambda$, for every $\lambda < \mu$. Then a graph lies in all these classes $\Gamma_\mu$ if and only if it contains a subdivision of the infinite binary tree.