Starting with and equivalence relation $\sim$ on sets, I have defined a partial order using union in a familiar way: $A \sqsubset B$ iff $A\cup B\sim B$.
My question: what property of $\sim$ is needed to make $A \sim B$ imply that $A \sqsubset B$?
Presently I use that "my" $\sim$ is "closed under union", as in: $A\sim B$ and $A'\sim B'$ imply $A\cup B \sim A'\cup B'$. But I suspect there might be some general result in order theory.