In Theorems in the form of "if and only if" such that the proof of one direction is extremely EASY to prove and the other one is extremely HARD, bof suggested the question in the title is more interesting, which motivates me to create this list. So my request is:

Theorems in the form of "if and only if" that the proofs of BOTH directions are nontrivial. The 'if and only if' formulation should be as natural as possible.

Thanks in advance.


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    $\begingroup$ I suppose the set theorists can provide some good examples of the form "ZF + X is consistent iff ZF + Y is consistent". For example, ZFC + Souslin's hypothesis is consistent iff ZFC + Borel's conjecture is consistent; but this example is kind of artificial. $\endgroup$ – bof Jan 11 at 9:05
  • $\begingroup$ A potentially even simpler example is "X is equivalent to Y in this set theory", e.g. all the ZF equivalents of the axiom of choice. But it may be hard to meet the OP's standard for the proof being "nontrivial". For example, the proof AC is equivalent to Zorn's lemma isn't that long. Maybe two equivalents of AC should be compared, e.g. Kőnig's & Zermelo's theorems. $\endgroup$ – J.G. Jan 13 at 8:30
  • $\begingroup$ @YuiToCheng comment of this form are usually not productive and thus routinely removed. If you want further information on this in general terms please ask in chat (Math Mods' office) or on meta. Preferably the former. $\endgroup$ – quid Jan 13 at 19:21

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