Strictly Increasing Mapping is Increasing

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

Proof
Let $$\left({S; \le_1}\right)$$ and $$\left({T; \le_2}\right)$$ be posets.

Let $$\phi: \left({S; \le_1}\right) \to \left({T; \le_2}\right)$$ be strictly increasing.

Note that, from Asymmetric Ordering, $$x \le_1 y \Longrightarrow x =_1 y \lor x <_1 y$$. So:

This leaves us with: