Why is a *-finite standard set a finite set?

Question

In internal set theory, why is it that if there is a bijection between $x$ standard and $n\in\mathbb{N}$, then $n$ is necessarily standard ?

Answer 1

In priciple you are asking wyh $\{1,2,3,...,\mu\}$ can only be a standard set if $\mu$ is standard?

By set operations, if $\{1,2,3,...,\mu\}$ is standard then also $\{mu+1,\mu+2,...\}$ must be standard. Since the minimum of a set of natural integers is a standard operation, then $\mu+1$ and thus $\mu$ must be standard.