No, the first death is not eternal, all are resurrected from it. At that time death is swallowed up in victory, and itself destroyed in the lake of fire. After that, the alternative for not being in the Lamb's book of life is the second death--something different than the first death, since "death" has already been swallowed up in victory and destroyed. Death has no more power over anyone. Therefore this second death is necessary for all who refuse to abide by the King's wishes, because they won't be welcome in the King's kingdom. But if the resurrected have immortal bodies now, then "their worm dies not". They have no escape.Symbolic for-- As each new day dawns in Gods kingdom, those destroyed in the lake of fire, miss each new day, miss each new day of Gods loving kindness, it never ends-likened to be eternal suffering. But a single teaching proves 100% it is symbolism= A place where the worm ( Maggot) dieth not----- No worm( maggot) could survive an hour in a hot firey place, proving its symbolic. Not to mention only a sadist could create an eternal suffering, one who would throw justice away--Thats not the true God-Deut 32:4-- only the blind guides who do not know God teach it is literal--The true God set-LIFE or DEATH before all-Deut 30:19--both are eternal.