This comes from Notes on Set Theory, 2nd edition.
Let $\leq$ be a total ordering of a set $A$ and define on the inductive poset $\mathcal{P}(A)$ the mapping
$$\pi(X) := \{y \in A | (\forall x<y)[x \in X] \}.$$
It can be verified that $\pi$ is monotone, and hence has unique strongly least fixed point $A_{\omega}$. Prove that
$$x \in A_{\omega} \iff M_x := \{(s,t) \in A \times A \space | \space s \leq t < x \} \text{ is a well ordering.}$$
By definition of $A_{\omega}$ we know that $A_{\omega}$ is the smallest set $B$ such that $(\forall y < x) [y \in B]$ implies $x \in B$.
I am not sure how to solve this problem. The difficulty here seems to be that $A$ need not be a well ordering (we cannot assume the axiom of choice here). I am not sure if I can prove whether $A_{\omega}$ is well orderable or not; that may make this problem a lot easier. Judging from the chapterthis question came from, it seems like the fact that a family of well ordered sets are well orderable and the property and exitence of Hartog's set may be used to prove this, but I am not sure how to apply those here.