Is it possible to lose HP by levelling up? If it is, is it possible to die by levelling up? [duplicate]

When you level up, you gain a number of Hit Points equal to 1d# (6, 8, 10, or 12) + your Constitution modifier. The lowest Constitution modifier possible is -4 (possible by rolling, and rolling 4 1s). If a PC with a negative Constitution modifier, negative by more than 1, levelled up, and rolled a 1 on the levelling up HP dice, would they lose HP? And, would it be possible to die by this HP loss?

For example, a Wizard with a 3 in Constitution, would have 2 HP at first level. At second level, we’ll say hey rolled a 1 on his Hit Points die. This would mean he gained -3 Hit Points. Would that kill him? I see no rules against this, but it seems to go opposite to the phrase "Levelling Up".