Prove the next equivalence:
(1) Axiom or Regularity
(2) If $X$ is a non-empty set, $X$ has an $\epsilon_x$-minimal element. This means, There exists a $z \in x$ such that $\lnot (x \in_x z )$ for all $x \in X.$
I got (1) $\rightarrow$ (2) wrong in an exam.
Because I didn't prove that "the $z$ existed." I am not being able to understand this observation, and so, I don't know how to correct this proof, or taking it into the right direction. So here's my attempt:
The Axiom of Regularity says: Let $A$ be a set, with $A \neq \emptyset $. There exists an $a \in X$ such that, if $b \in a$, then $b$ is not in $A$.
Proof. As $X$ is not empty, there exists a $z\in X$. If $z\in X, $ and for some $x$, $x\in X$ then, by the Axiom of Regularity, $x \notin_x z.$ So, $\lnot(x \in_x z).$