In textbooks, supremum is often defined as follows:
Let be a partially ordered set, and let .
(1) is a least element of if for all .
(2) is an upper bound of if for all .
(3) is called a supremum of if it is the least element of all upper bounds of . Denote it as .
If we express these definitions in wffs of first-order logic, then
(1) is a least element of if the following wff is true: .
(2) is an upper bound of if the following wff is true: .
Now if is an empty set , it follows that any element of is an upper bound of , for in (2) the left part of is always false, so the implication is vacuously true, and the whole quantifier expression is true. This is like what we do in proving that the empty set is contained in any set. Therefore, by definition, is the least element of the set consisting of all upper bounds of , that is, just the least element of . I'm not sure if it is right or not, does anybody know? Thanks.