Woset_is_Isomorphic_to_Set_of_its_Initial_Segments

Let $\struct {S, \preceq}$ be a well-ordered set. Let $A = \set {a^\prec: a \in S}$ where $a^\prec$ is the strict lower closure of $S$ determined by $a$. Then $\struct {S, \preceq} \cong \struct {A, \subseteq}$ where $\cong$ denotes order isomorphism.