Uniqueness of $0$

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I am just a high school student. So, I don't know much about higer mathematics. But a question has come to my mind.

Question:

$$0 = 0^{n-(n-1) } = \frac {0^{n}}{0^{n-1}} = \frac {0}{0}$$

But $\frac {0}{0}$ is undefined. So, this means $0$ is undefined. But this is totally wrong.

I want to know are there any axioms, theorems or arguments claiming that it's wrong?