None of the intervals can be unbounded, right or left rays. Otherwise there is a counter-example.

Name the interval with the least left end point . There are at most six intervals that may share a point with it. So we have at least 43 which have no point common with . Of those, name the interval with the least left end point . Again are at most six intervals that may share a point with , leaving 36 that have no point either of the two named. We continue getting from 28 and so fourth until we get from 8.

Now you have 8 intervals that are pairwise disjoint.