(Why) are $0, 1, 10, 120, 1540, 7140$ the only triangular tetrahedral numbers?

Question

I recently came across entry A027568 of the OEIS, which reports the numbers $$0, 1, 10, 120, 1540, 7140$$ as the set of integers which are both triangular and tetrahedral; that is, each can be written as both $\binom{n+1}{2}$ and $\binom{m+2}{3}$ for integers $m,n$. But the entry is not explicit about whether these are the only such integers or merely the only known examples. So, the question: Are these known to be the only triangular tetrahedral numbers, and if so how?