Given a chain-complete poset $P$, every $x\in P$ lies below some maximal element.
Every inductive poset has
enough maximal elementsa maximal element.
Chain-complete means every chain has a least upper bound, inductive means every chain has an upper bound.
Are the above statements both correct and equivalent? What exactly does "enough maximal elements" mean? Can one proof $1\implies 2$ without any further assumptions?
/edit: The "enough" thing is from handwritten notes maybe it's a mistake. "All chains have upperbounds, then there is a maximal element" seems to be the most common definition. If we take this as $2$, are $1$ and $2$ equivalent? If I'm not mistaken, $1$ has both a stronger condition and a stronger conclusion?