Your proof system is neither sound nor complete, since $A$ is not a true proposition unless $A \equiv \top$, In which case $\lnot A \equiv \bot$ is not a true proposition. This argument shows that every sound proof system is also consistent.

Your proof system is neither sound nor consistent, since $A$ is not a true proposition unless $A \equiv \top$, In which case $\lnot A \equiv \bot$ is not a true proposition. This argument shows that every sound proof system is also consistent.