Not quite.
Adam did die when he ate the forbidden fruit.
Yet, it was not physical death.
It was spiritual death.
When he ate he ceased functioning as body, soul, and human spirit.
He became in function only body and soul.
"That which is born of the flesh is flesh;
and that which is born of the Spirit is spirit." John 3:6
Unbelievers need to become born again.
That is why Jesus told Nicodemus he needed to be born again...
Telling him, that Spirit begets spirit.
"That which is born of the flesh is flesh;
and that which is born of the Spirit is spirit."
In that manner a man becomes what God always intended a man to be.
Body, soul, and human spirit.
Now in Scripture there is also a spirit of man spoken of.
Its usage is not spoken in reference to the human spirit, which is involved in spiritual matters.
Its usage has a different meaning.
More in line with manifesting the likes of a team having "school spirit."
grace and peace .......