Lower Sections of Well-Ordered Classes are Order Isomorphic at most Uniquely

Theorem

Let $\struct {A, \preccurlyeq_A}$ and $\struct {B, \preccurlyeq_B}$ be well-ordered classes.

Let $L_A$ be a lower section of $\struct {A, \preccurlyeq_A}$.

Then:

$(1): \quad L_A$ is order isomorphic to at most one lower section $L_B$ of $\struct {B, \preccurlyeq_B}$

$(2): \quad$ If such an $L_B$ exists, there exists exactly one order isomorphism from $L_A$ to $L_B$.

Proof

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 2$ Isomorphisms of well orderings: Corollary $2.7$