Having already proved $A\vdash B$, $B\vdash C$ and $\neg C$, is this a fine proof proving $\vdash\neg A$?

Question

since $A\vdash B$ is true $A\rightarrow B$ is also true.

since $B\vdash C$ is true $B\rightarrow C$ is also true.

1. $\quad\bullet$
2. $\quad\bullet\quad\bullet\; A$ --- assumption
3. $\quad\bullet\quad\bullet\; A\rightarrow B$ --- Theorem Intro
4. $\quad\bullet\quad\bullet\; B$ --- By $\rightarrow$ Elim 2,3
5. $\quad\bullet\quad\bullet\; B\rightarrow C$ --- Theorem Intro
6. $\quad\bullet\quad\bullet\; C$ --- By $\rightarrow$ Elim 4,5
7. $\quad\bullet\quad\bullet\; \neg C$ --- Theorem Intro
8. $\quad\bullet\quad\bullet\; \bot$ --- $\bot$ Intro 6,7
9. $\quad\bullet\; \neg A$ --- $\neg$ Intro 2 $-$ 8