You need the Cantor-Bernstein in sets because of how you define the "equality" or the order of sets.
A set A has greater than or equal cardinality than another set B iff there exists an injection from B into A.
two sets have equal cardinality iff there exists a bijection between the 2 sets.
so to prove the equality of cardinalities, you have to establish the existence of a bijection.
I think the order axioms result from the definitions of the reals as an ordered fields, and the theorem than can be proved from the ordered axioms is that a number is either positive, negative or 0.
from Mathworld
http://mathworld.wolfram.com/TrichotomyLaw.html
Quote:
Every real number is negative, 0, or positive. The law is sometimes stated as "For arbitrary real numbers a and b, exactly one of the relations a < b, a = b, a > b holds" (Apostol 1967, p. 20).
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I'm guessing that you have to take either one of them as an axiom, and the other automatically follows...