Condensation bounds a constructible real

Choose a nonzero limit $\theta$ with $x,\omega \in L_{\theta }$ and form $H={\mathrm{Hull}}^{L_{\theta }}(\omega \cup \{\omega ,x\})$. F1 gives $H\prec L_{\theta }$ and a bijection between $H$ and $\omega$. In particular every natural number belongs to $H$, not merely the set $\omega$ as one element.

Collapse $H$ by $\pi$ to $M=L_{\beta }$ using F2. Since $\omega \subseteq H$ is transitive, F3 fixes every natural number. The map $\pi$ is an isomorphism onto the transitive set $M$: if $y\in \pi (x)$, surjectivity gives $u\in H$ with $\pi (u)=y$, and relation reflection gives $u\in x$; conversely $u\in x$ implies $\pi (u)\in \pi (x)$. Since $x\subseteq \omega \subseteq H$ and $\pi (u)=u$ there, $\pi (x)=\{\pi (u):u\in x\}=x$. Hence $x\in M=L_{\beta }$. This includes the empty real.

The collapse is a bijection, so $M$ is countable. Since $\beta =\mathrm{Ord}\cap M\subseteq M$, the ordinal $\beta$ is countable and therefore $\beta <{\omega }_1$. Thus the hull calculation proves the claimed bound and, consistently with F4, yields $x\in L_{{\omega }_1}$. No ambient Choice is used.