Infection requires virus particles to be physically present in transmission vectors (fluids, droplets, etc). There's generally also a dose-response effect where more particles means more chance of evading the immune system well enough to establish replication in the victim.
So a lack of anything to detect, means a lack of anything to spread an infection.
One should add that for that the threshold for detection has to be lower than the threshold for infection.
Example: let the detection threshold be 10 particles/ml and the infection threshold be 100 particles/ml (*) -> then undetectable implies that it is very improbable that an infection will take place.
(*) This is a very crude description. Think of it like this: Every single virion (virus particle) has a very low probability of causing an infection itself but there is a high number of them and for one them it might just work out (higher viral load -> higher risk of successful infection)
This statement is logical and makes perfect sense - but it's narrower than the original one.