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That is not correct, only those who have faith in Jesus Christ as their Lord and Savior and follow Him have their name written in the Book of Life.
Everyone who has faith in the lord will follow him,, there is no if about it. They are the ones who obey the commands of God.
Unbelievers do not have their name written in that book. To say everybody's name is written is a false theology, only believers names are written and to be blotted out of that book you would have to walk in disobedience to Him.
yet that is what you are proposing.
He gave the Pharisees and the rich man warnings because how they walk in life is for self and not for God. They do everything for themselves and if they do not repent and start living for God then they will find out their name was never written in the Book of Life do to their disobedience.