# Proper Subtower is Initial Segment

Jump to navigation Jump to search

## Theorem

Let $\left({T_1,\preccurlyeq}\right)$ be a proper subtower of $\left({T_2,\preccurlyeq}\right)$.

Then $\left({T_1,\preccurlyeq}\right)$ is an initial segment of $\left({T_2,\preccurlyeq}\right)$.

## Proof

Define the set:

$Y = \left\{ { y \in T_1: S_y \text{ is an initial segment of } \left({T_2,\preccurlyeq}\right) } \right\}$.

Then:

 $\displaystyle S_x \left({T_1}\right)$ $=$ $\displaystyle \left\{ {b \in T_1, x \in T_1: b \prec x}\right\}$ Definition of Initial Segment $\displaystyle$ $=$ $\displaystyle \left\{ {b \in T_2, x \in T_2: b \prec x}\right\}$ Definition of Proper Subtower in Set, as $T_1 \subseteq T_2$ $\displaystyle$ $=$ $\displaystyle S_x \left({T_2}\right)$ Definition of Initial Segment

By induction on well-ordered sets, $Y = T_1$.

That is, $\left({T_1,\preccurlyeq}\right)$ is an initial segment in $\left({T_2,\preccurlyeq}\right)$.

$\blacksquare$