Relation Segment is Increasing

Theorem

Let $S$ be a set.

Let $\mathcal R, \mathcal Q$ be relation on $S$ such that

$\mathcal R \subseteq \mathcal Q$

Let $x \in S$.

Then

$x^{\mathcal R} \subseteq x^{\mathcal Q}$

where $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$.

Proof

Let $y \in x^{\mathcal R}$

By definition of $\mathcal R$-segment:

$\left({y, x}\right) \in \mathcal R$

By definition of subset:

$\left({y, x}\right) \in \mathcal Q$

Thus by definition of $\mathcal Q$-segment:

$y \in x^{\mathcal Q}$

$\blacksquare$