## What we learn in set theory :)

In general "<" equivalent to $\in$ is not the case, but the specific way $S_y$ is defined make it true, since whenever x \in S_y, it must be the case that x < y, so x_{n+1} = f(x_n) \in S_{x_n} means x_{n+1} < x_n.

Since we just have x_{n+1} \in S_{x_n}, we are still good regarding the Foundation Axiom of ZFC. If on the other hand, we have x_{n+1} \in x_n for all n, then the theorem rules it out.

Since we just have x_{n+1} \in S_{x_n}, we are still good regarding the Foundation Axiom of ZFC. If on the other hand, we have x_{n+1} \in x_n for all n, then the theorem rules it out.

(*) ( \forall x < y P(x) ) => P(y)

When y is the <-least element in A, the LHS of the (*) "vacuously" holds, meaning, for all elements in the empty set, all property holds, including this P, thus RHS holds. Therefore P(y) holds for free.

When y is the <-least element in A, the LHS of the (*) "vacuously" holds, meaning, for all elements in the empty set, all property holds, including this P, thus RHS holds. Therefore P(y) holds for free.