How to prove that $\text{pred}_{(A,\prec_A)}(y)=\{x\in A \,|\, x\prec_A y\}$

Question

In Schimmerling's book on set theory, question 3.1 reads:

"Let $(A,\prec_A)$ be a wellordering such that $A\neq\emptyset$. for each $y$ define $\text{pred}_{(A,\prec_A)}(y)=\{x\in A \,|\, x\prec_A y\}$.

Suppose that $S\subsetneq A$ and, for all $x,y\in A$, if $y\in S$ and $x\prec_A y$ then $x\in S$.

Prove that there exists $y\in A$ such that $S=\text{pred}_{(A,\prec_A)}(y)$."

I have been trying for days to find a solution to this problem, and would appreciate any hints.

Answer 1

Hint: Since $S$ is a proper subset of $A$, let $y=\min A\setminus S$. Show that this is the $y$ you're looking for.