I tried to prove in a natural deduction style the basic ( almost trivial maybe ) theorem on well ordered sets : " Any well ordered set is totally ordered".
Would you please tell me which objections could be made to this proof. Any comment is welcome.
I think I used " consructive dilemma" in the subordinate derivation. Is this correct?
Which justification could I bring to : " a and be belong to A, therefore {a,b} is a subset of A" ? ( Is a set theoretic axiom required here?)
There is nothing wrong with the proof.
You need the axiom of pairing to form the set $\{a,b\}$.