The problem is we can only "prove" this by recourse to economic models. Historically, societies that had slavery have been more advanced than those without most of the time, but of course, slavery (that is, chattel slavery, not other forms of unfreedom) requires things like organised military force, a legal system, a market economy and the wealth surplus to sustain slaveowners and luxury slaves, so that's probably not surprising. Non-slave societies through much of history havve been those without the social infrastructure to sustain slavery.
Now, I know the argument about Greco-Roman slavery, but I don't buy it. Hellenistic and Roman times saw slavery expand from a mainly luxury aspect of the urban market to a mainstay of the economy, but also saw an explosion of technology. The eighth through eleventh centuries in Europe saw a rise in technology use that coincided with a huge growth industry - slaving. Renaissance Italy had slaves aplenty while contemporary Muscovy did not. Of course it's not that slavery drives technology, either. The number of slaves held in Europe after around 1200 seems to drop precipitously, but technology moves on.
I suspect there is a different thing at work here. Slavery is a very effective form of securing a supply of labour, but it has its limits. Once a society finds different ways of securing labour (be it through sustaining a population surplus, through legal bonds, social conditioning, a high degree of division of labour, or the promise of affluence), the comparative advantage of slavery drops. It stays attractive, but only in certain fields (you can still run some industries effectively on slave labour, as we see happening, and at the individual level, owning a slave beats the hell out of owning a Ferrari for ego gratification, apparently). In the Western world, slavery also became the foil against which tro define its predominant ideology of individual liberty, but I don't think that this is a necessary precondition for modernity. You could have highly complex, industrialised societies that practice slavery. in fact, we have those societies, they just don't admit to it for purely ideological reasons.
As to the Roman steam engine - that didn't exist. And the anecdote about Tiberius is exactly that, an anecdote, and one that does not mention slaves. What the Roman world severely lacked to progress technologically beyond its (pretty damned impressive) level was a concept of modern economics. That slavery can be integrated into one very efficiently is witnessed by Early Modern America.