let X = {the unbelieving} and let Y = {the believing}
proposition: X ∩ Y = ∅
proof:
suppose ∃k: k ∈ X ∩ Y
then k both believes and does not believe, and by John 3:18 k is both not condemned and condemned already.
by James 1:7-8, k receives nothing from the Lord.
by John 3:14-16, k receives eternal life.
therefore k has eternal life, but not really.
it follows that k does not believe, by 1 John 5:10-13, so k ∉ Y
this implies k ∉ X ∩ Y, therefore X ∩ Y = ∅
QED
therefore there must be some transformation G:X → Y or else there is no salvation.