For any non empty set $X$; $A,B \in \mathcal{P}(X)$, the power set, define $A\sim B \iff $ both $A$ and $B$ have same cardinality and let $\tilde{X}$ denote the equivalence classes.
For $\tilde{A}, \tilde{B}\in\tilde{X}$, define $\tilde{A} \preccurlyeq\tilde{B} \iff |\tilde{A}| \leq |\tilde{B}|$ where $|\tilde{A}|$ denotes the cardinality of $\tilde{A}$.
Now, we know that the axiom of choice implies that $\preccurlyeq$ is a total order. What I wish to know is whether it is a well-order.