I need to prove that given $B \in \omega$ ,meaning $B$ is finite ordinal and $B \not= \emptyset $ then
$\exists A \in B \forall x \in B (A=x , x \in A) $
I know that because $B$ is ordinal then by definition of ordinal for every two sets $x,y$ in $B$ one of the following is true $x=y , x \in y , y \in x$
But i did think only in induction to solve it, please help with the proof.
Hint: If $x$ is the largest element of $\alpha$, then $s(x)$ is the largest element of $s(\alpha)$.
You can use this to construct the inductive step. The base step is easy since $\{0\}$ contains only one element.