Claims About a Difference-Closed Sequence

A five-term sequence $a_1,a_2,a_3,a_4,a_5$ satisfies $0\leqslant a_1<a_2<a_3<a_4<a_5$, and for all $i,j$ with $1\leqslant i\leqslant j\leqslant 5$, the difference $a_j-a_i$ is a term of the sequence. Determine, with proof, which of the following claims are true.

1. $a_1=0$. 2. $a_5=4a_2$. 3. $\{a_n\}$ is an arithmetic progression. 4. The set $A=\{a_i+a_j\mid 1\leqslant i\leqslant j\leqslant 5\}$ has $9$ elements.