Infinite Sequence Property of Well-Founded Relation/Forward Implication

Theorem

Let $\struct {S, \RR}$ be a relational structure.

Then there exists no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:

$\forall n \in \N: \paren {a_{n + 1} \mathrel \RR a_n} \text { and } \paren {a_{n + 1} \ne a_n}$

Aiming for a contradiction, suppose there exists an infinite sequence $\sequence {a_n}$ in $S$ such that:

$\forall n \in \N: \paren {a_{n + 1} \mathrel \RR a_n} \text { and } \paren {a_{n + 1} \ne a_n}$

Let $T = \set {a_0, a_1, a_2, \ldots}$.

Let $a_k \in T$ be a minimal element of $T$.

That is:

$\forall y \in T \setminus \set {a_k}: \neg \paren {y \mathrel \RR a_k}$

But we have that:

$a_{k + 1} \mathrel \RR a_k$ and $a_{k + 1} \ne a_k$.

So $a_k$ is not a minimal element.

It follows by Proof by Contradiction that such an infinite sequence cannot exist.

$\blacksquare$