Perfect Subsets

For $n \in \mathbb{N}^*$, write $|A|$ for the number of elements of a set $A$ and $\min(A)$ for its smallest element. A nonempty set $A \subseteq \{1, 2, \ldots, n\}$ is called an $n$-perfect set if $|A| \leqslant \min(A)$. Let $a_n$ be the number of $n$-perfect sets. Determine, with proof, which of the following are true:

1. $a_4 = 7$; 2. adding $1$ to every element of an $n$-perfect set always yields an $(n+1)$-perfect set; 3. the number of $(n+2)$-perfect sets $A$ with $|A| > 1$ and $n + 2 \in A$ is $a_{n+1} - n$; 4. the number of $(n+2)$-perfect sets $A$ with $|A| > 1$ and $n + 2 \notin A$ is $a_{n+1} - n - 1$.