Suppose following axiomatization of real numbers:
( is syntactical sugar for uniqueness quantifier)
(< 1) forall forall , one of the following holds: , ,
(sup) Let be non-empty set, s.t. it has upper bound. Then
Now following theorem holds: There is precisely one element (which will be denoted ), which is solution of equation .
Proof: Assume some fixed . Let the only solution of be denoted by symbol . So holds.
Now proof is straightforward and I will not finish it. My problem is that is formal symbol of our first-order theory (i.e. it is non-logical constant) and no axiom define any property that should obbey. However the first step of the theorem says something like . My question is why is this correct? I don't see any inference rule which interlinks and axiom (+ 3).
any ideas why is that correct? thanks