Strictly Increasing Mapping is Increasing

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

Proof
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be strictly increasing.

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

So:

This leaves us with:

Also see

 * Strictly Decreasing Mapping is Decreasing