Class of ordinals is defined as follows:
- If $A$ is 'any' well-ordered set, there exists an element $a$ in the class such that $A$ is similar to $a$.
- If $A$ is a well-ordered set and $a$,$b$ are elements in the class, then [$A$ is similar to $a$ and $A$ is similar to $b$ ⇒ $a=b$].
I have constructed a well-ordered class, $C$, which satisfies two conditions and is $\in$-well-ordered.
First, I have proved that if there exists another class, $D$, satisfying above two conditions, $D$ is well ordered by -< (isomorphism).
Second, I have proved that there exists an isomophism $f$ between $C$ and $D$.
My questions is if $\{r_i\}$ is a family of ordinals in $C$, $f(\bigcup r_i$) = $\bigcup f(r_i)$?
If this is false, how come $sup\{r_i|i \in I\}$ = $\bigcup r_i$ is generally true?