There is a least set containing all elements of any given set. More can be said. EFA which has axioms for finite sets of nonnegative integers. Every nonempty set has an epsilon minimal and an epsilon maximal element. Separation for bounded formulas.
Every set is in one-one correspondence with an ordinal.
And again we can optionally add 8. There is a least set containing all elements of any given set. Every nonempty set has an epsilon minimal and an epsilon maximal element. Separation for bounded formulas. Every set is in one-one correspondence with an ordinal.