This is definitely not true. One obvious example is that important scientific work is done turning unfalsifiable theories into falsifiable ones. After gravitational waves were predicted by general relativity there was some argument about whether they were a real observable thing or just an artifact of the maths. Those guys were undoubtedly doing "real science" while discussing something that some of them thought was undetectable (and therefore unfalsifiable).
Another example is that a lot of science is exploratory without any particular theory you're trying to falsify.
Science is more interesting and broad than your dogma about what it should be.
The vast majority of the time, when a working scientists says "proved" they understand the epistemic limitations of the terminology.
What is your source? Wikipedia disagrees with you.
https://en.m.wikipedia.org/wiki/Falsifiability
TL;DR: If a theory isn’t falsifiable—if it doesn’t make any predictions which can be proved or disproved—then it’s more philosophy than science. By this standard, for example, string theory fails.