In Introduction to Set Theory by J. Donald Monk, he defines ordinal as follows.
Definition (1): $A$ is an ordinal iff $A$ is $\in$-transitive and each member of $A$ is $\in$-transitive. $A$ is $\in$-transitive iff for all $x$ and $y$ , $x \in y \in A \implies x \in A$.
And using definition (1), I have a problem of showing $0$ is an ordinal.
My solution: Let $x \in y \in 0$, then show that $x \in 0$, but in the theorem shown before this, there cannot exist an $x \in 0$ for any $x$, for if $x \in 0$, then $x$ is a set and $x \neq x$, which is absurd. Hence there cannot be an $x$ such that $x\in 0$ .But if this is the case, we cannot shown the above theorem by definition. Any ideas?
thanks for your help.
Edit: Initially I have 2 question to ask. But later on, I have found the answer to my first question. That is why I deleted it and the question post above is my second question.