= Uncountable transitive set model has uncountable ordinal height
Let $M$ be a transitive set model of <ZFC>. If $\operatorname{Ord}\cap M$ were countable, then every $x\in M$ would be countable: the internal <axiom of choice> supplies a bijection from $x$ to an ordinal of $M$, which is externally countable. For every $\alpha\in\operatorname{Ord}\cap M$, the internal rank $V_\alpha^M$ is an element of $M$ and hence countable. The <Axiom schema of replacement> inside $M$ gives $M=\bigcup_{\alpha\in\operatorname{Ord}\cap M}V_\alpha^M$, a countable union of countable sets. Thus every uncountable transitive set model has uncountably many ordinals.
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