Strictly Increasing Mapping is Increasing

Theorem
A mapping that is strictly increasing is an increasing mapping.

Proof
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be strictly increasing.

From Strictly Precedes is Strict Ordering:
 * $x \mathop {\preceq_1} y \implies x = y \lor x \mathop {\prec_1} y$

So:

This leaves us with:

Also see

 * Strictly Decreasing Mapping is Decreasing