Let's say we have a universe $U$ of $n$ elements and a collection $S$ of $m$ subsets of $U$, i.e., $S=\{S_1,\ldots,S_m\}$, and a positive integer $k$. If I ask is there a set cover of $U$ of size $k$ or less, then this is NP-complete. Now suppose that I add the following: $|S_1| \gt |S_2| \gt \ldots \gt |S_m|$.Is this still NP-complete?I guess that, given 1., we know that the sets $S_i$ are all distinct and then the only cover for $U$ is $S$ but I am not sure about this.          
Is this variant of set cover NP-complete?
np complete;set cover
As quicksort comments, you can convert any set cover instance to your special case by adding dummy elements. Take a large enough pool of dummy elements, and put all of it into a new set $S_1$. Then add subsets of $S_1$ to the other sets to satisfy your cardinality condition, taking care to leave some of them out. Every solution has to take $S_1$, and otherwise it just has to be a solution to the original problem.This is of course not a formal reduction - this is left to the reader.