> Moreover, mathematicians rarely prove anything from axioms, they start from other statements which are considered more trivial.
Not more trivial but from the ones already proved and those closer to the thing you are proving. There is no need to go back to the axioms if you know, and can reference, proof of step N-1, just go from there.
There is also this 'misconception' that mathematical theorems follow from the axioms. They do, of course, but the axioms were choosen just right to make things that were working to still work, with some weird consequences like axiom of choice
Not more trivial but from the ones already proved and those closer to the thing you are proving. There is no need to go back to the axioms if you know, and can reference, proof of step N-1, just go from there.
There is also this 'misconception' that mathematical theorems follow from the axioms. They do, of course, but the axioms were choosen just right to make things that were working to still work, with some weird consequences like axiom of choice