Understand that the spirit does not die, unlike the body. Stopping the body from existing does not erase the spirit, it only separates it from the body.
Also, the spirit of man is different from the Spirit of God. However, there's a measure of the spirit of the Spirit of God in him, the spirit of man could be overwhelmed and overcharged and overshadowed by the Spirit of God. The spirit of man could be referred to as the conscience of man, the mind of man, the inner voice of man, the consciousness of man.
Let me give you this illustration, the spirit of man stands between the body and the soul. It is like a messenger. It takes messages from the body, processes it, displays it in his subconsciousness, and finally dumps it with the soul. In other words, the soul (though I will talk more about it in the next chapter) is the recipient of every action performs by the body, through the control of the perverted senses and tarnished spirit.
IS THE SPIRIT OF MAN THE SPIRIT OF GOD?
Let's see a couple of verses:
"The spirit of man is the candle of the LORD, searching all the inward parts of the belly. (Proverbs 20:27)"
Again,
"But there is a spirit in man: and the inspiration of the Almighty giveth them understanding. (Job 32:8)"
From the above, and like I said at the beginning, it's clear that these are two different personalities here: The spirit in man (or the spirit of man), and the Spirit of God. However, the spirit of man is a part of God as well (the candle/lamp, or conscience) searching the inward parts, making sure wrongs and corrections are detected and made, while the Spirit of God is the fullness of the Personality of God, who is also the Third Person in the Trinity, and this is God Himself in the Spirit form.
On the inside of man, when he was initially created, it was just the spirit of man and that of God that were in fellowship. Man's spirit was always and totally in agreement with that of God, as there was no contrary suggestions from any third party, there wasn't strive or resistance, until Satan came and another thought (spirit) came as the third party, bringing about confusion, doubts, faithlessness, impurity and fight in the spirit of man against the Spirit of God. The spirit of man began to have doubts about God whom it already had no issues with initially; next, it eventually began to exhibit faithlessness in the power of God to do all things; and pollution of filthiness evolved, creating dispute between these two friends who never had an issue (man's spirit and God's Spirit). This is the reason why, when tempted, the spirit of man, halt between two opinions, whether to succumb or not. Before now, there was nothing like being tempted, and if there was, the chances for falling was zero, because the door was sealed against this third party. But as soon as man gave in to the suggestion from the serpent (devil), he opened the door for him (devil), which gave him the license to always interfere in the affairs between man and his Creator.
STRIVE INSIDE
Let's remind ourselves with what God said in Genesis 6:3, "And the LORD said, My spirit shall not always strive with man, for that he also is flesh: yet his days shall be an hundred and twenty years." Understand something here, actually, the days of man was reduced here. You would agree with me that it's almost always impossible for anyone to live above 120 years in the world today. The Spirit of God had had enough of man's stubbornness. He can't keep contending with him for so long like before. Think about someone like Methuselah who lived 969 years. How do we expect the Spirit to keep striving with man's for so long? Not that God's Spirit gets weary like ours, but all that would be unnecessary. Therefore, the lifespan of man was reduced.
Like I mentioned, this contention on the inside of man was never there, until he (man) allowed the involvement of the serpent at the garden. Else, service or fellowship would have remained the only thing going on in the heart of man even till this very day.