Just a curiosity:



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> Is there an assertion of which a proof (formalizable, say, in ZFC) is not known but a proof that it's *not* undecidable (in ZFC) *is* known?
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Edit: after the comments, I think the actual question was 



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> Is there an ("interesting") assertion of which neither a proof (formalizable, say, in ZFC) of it or its negation is known but a proof that it's *not* undecidable (in ZFC) *is* known?
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Is there an "undecided" assertion of which a proof that it's not undecidable is known?
If it's known that some statement $S$ is decidable in ZFC, then you can just run a computer program that enumerates all ZFC-proofs and stops when it finds a proof of $S$ or a proof of $\neg S$. By hypothesis, this algorithm is guaranteed to terminate. Therefore, the only possible obstacle separating decidable statements from decided ones is computational complexity.


In other words, the only possible instances of what you're looking for are statements that have already been proved *up to a finite computation*. Until they were actually proved, the Kepler Conjecture and Catalan's Conjecture were perhaps the most interesting examples of this type. I can't think of other examples of comparable interest offhand, but maybe others can.


