This question stems from https://mathoverflow.net/a/6594/22332 and really is summarized in the title: Has there been any research on the power of $|x|\lt|y| \rightarrow |2^x|\lt|2^y|$? It seems like it would be an obvious candidate for extending ZFC by "intuitive" axioms, so I'd assume there was research and it was seen to not hold much power, but even then I'd be curious about those results!
Edit to add to save some link-clicking: The cited answer states that the statement is independent of ZFC (and is, obviously, confirmed by GCH) and its negation follows from "Martin's Axiom, when CH fails."