The total number of all different integers in all partitions of n with smallest part $\geq 2$

Question

I want to show that the total number of all different integers in all partitions of $n$ with smallest part $ \geq 2$ is $p(n-2)$.

Example: partitions of $7$ with smallest part $ \geq 2$ are (7), (5,2), (4,3), (3, 2, 2). Let’s count distinct parts in each, so we get respectively 1 + 2 + 2 + 2 = 7 = p(7 - 2) = p(5).

Could you give any reference to the proof of that fact? Or maybe some hints to prove it in laconic way?

Answer 1

For the sake of answering this question, answers are given over at https://mathoverflow.net/q/451202/20598.