# Order Isomorphism Preserves Initial Segments

## Theorem

Let $A_1$ and $A_2$ be classes.

Let $\prec_1$ and $\prec_2$ be strict orderings.

Let $\phi : A_1 \to A_2$ create an order isomorphism between $\left({A_1, \prec_1}\right)$ and $\left({A_2, \prec_2}\right)$.

Suppose $x \in A_1$.

Then $\phi$ maps the $\prec_1$-initial segment of $x$ to the $\prec_2$-initial segment of $\phi \left({x}\right)$.

## Proof

$\phi$ maps the $\prec_1$-initial segment of $x$ to:

 $\displaystyle \phi \left[{\left\{y \in A: y \prec_1 x \right\} }\right]$ $=$ $\displaystyle \phi \left[{\left\{y \in A: \phi \left({y}\right) \prec_2 \phi \left({x}\right) \right\} }\right]$ by the definition of initial segment and order isomorphism $\displaystyle$ $=$ $\displaystyle \left\{ \phi\left({y}\right) \in \phi \left[{A}\right]: \phi \left({y}\right) \prec_2 \phi \left({x}\right) \right\}$ by the definition of order isomorphism and image

This is the $\prec_2$-initial segment of $\phi \left({x}\right)$.

$\blacksquare$