I am a bit confused about constructing a Grothendieck topology from a Grothendieck pretopology, largely because I have a discrepancy in definitions of the former.
According to all of the questions I've seen on this site (and Wikipedia), there are three conditions for an assignment of sieves on an object to each object in a category to be a Grothendieck topology, namely stability under pullbacks, transitivity, and containing the maximal sieve.
nLab, however, also has the requirement that two sieves are covering if and only if their intersection covers, and mentions that 'here the saturation condition is important'. One direction of this hypothesis follows straight away if we are trying to show that we can construct a topology from a pretopology, but I don't see how, if we have two sieves $S,T$ which both contain a covering family, the intersection $S\cap T$ has to contain a covering family.
Could anybody please explain why the nLab has this condition, what it means by its comment, and what I'm missing?
Edit: Apparently (from the comments) the intersection condition is redundant. Does anybody know where I could find a proof of this to cite? I'm working on a tight word limit and would quite like to avoid having to state a proof myself if it's not just a quick one-liner...