Why such $\theta$ exists by replacement?

Question

In Jech's textbook set theory, theorem 2.27, the author is proving any well-founded relation has a height. As the picture below,

I don't understand this step. Why such $\theta$ exist?

Answer 1

Consider the function $x\mapsto\min\{\alpha\mid x\in P_\alpha\}$. Since $P$ is a set, its image is a set of ordinals. So there is some $\theta$ which is an upper bound.

Now it is easy to see that $P_\theta=P_{\theta+1}$, since no new members could be added, we have exhausted $P$.