How do you prove it was their spirit that died, since Scripture says it is the spirit that gives life? If the spirit in them died, how did their body continue to be physically alive since they were not born again? It was through the fallen, not dead, spirit within them, giving them physical life, that they knew they had sinned against God. Because they chose evil rather than good, the serpent spoke truth when he said "Ye shall not surely die." But that was only truth in part, the serpent knew that man having knowledge of evil would not physically die on the same one day they disobeyed God, but on that day (one) during the age in which they lived they became dead in trespasses and sins. After the fall, apart from the Spirit of Christ in them, man would remain dead to righteousness, goodness. That means man could never muster up faith to save themselves, being destined to die in their sins. Because apart from Christ none are good, none are righteous. The spirit in them did not die, rather the spirit giving man physical life became altogether without the ability to attain eternal life. It is in this manner that man died during the day/age/time in which A&E lived. Dead to God's Spirit, but very much alive to the spirit that works in the hearts of fallen man.
I didn't say the spirit died in the sense that it did not exist--I said they died spiritually in relationship to their fellowship with God. Notice what SCRIPTURE says about "spirit" death:
Revelation 3:1
“To the angel of the church in Sardis write: He who has the seven Spirits of God and the seven stars, says this: ‘I know your deeds, that you have a name that you are alive, but YOU ARE DEAD..
Ephesians 2:1
And you were DEAD in your trespasses and sins,
Ephesians 2:5
even when we were DEAD in our transgressions, made us alive together with Christ (by grace you have been saved),
Colossians 2:13
When you were DEAD in your transgressions and the uncircumcision of your flesh, He made you alive together with Him, having forgiven us all our transgressions,
Matthew 8:22
But Jesus *said to him, “Follow Me, and allow the DEAD to bury their own dead.
Hosea 13:1
When Ephraim spoke, there was trembling.
He exalted himself in Israel,
But through Baal he did wrong and
DIED [not literally]
1 John 3:14
We know that we have passed out of DEATH into life, because we love the brethren. He who does not love abides in death.
Romans 7:9
I was once alive apart from the Law; but when the commandment came, sin became alive and I DIED; [not literally]
rwb: Will you plealse explain how Adam and Eve could have their eyes open "in the day" (immediately) but didn't die in that same day? It's a dilemma for you.