Proof of Decidability of Monadic First Order Logic

I have no reference, but it seems to me fairly straightforward that monadic FOL would be decidable: If you only have monadic predicates, and given that any sentence in monadic FOL would have only a finite number of monadic predicates (say $n$), then you can distinguish at most $2^n$ different 'kinds' of objects in terms of them having or not having the property as expressed by the predicate for each of the $n$ predicates. And without being able to express identity (which requires an at least 2-place relation), you cannot express that there are at least two of a certain 'kind'. Ergo: if there are no models for a sentence with $2^n$ objects in its domain, then there are no models either with more than $2^n$ objects in its domain. So, to see whether some sentence is a monadic FOL valid sentence, just check see if there is a model for its negation with $2^n$ objects: if ao, then the sentence is not valid, but if not, then it is.