Min Operation is Commutative

Theorem
The Min operation is commutative:


 * $\map \min {x, y} = \map \min {y, x}$

Proof
To simplify our notation:
 * Let $\map \min {x, y}$ be (temporarily) denoted $x \underline \vee y$.

There are three cases to consider:


 * $(1): \quad x \le y$
 * $(2): \quad y \le x$
 * $(3): \quad x = y$

$(1): \quad$ Let $x \le y$.

Then:

$(2): \quad$ Let $y \le x$.

Then:

$(3): \quad$ Let $x = y$.

Then:

Thus $\underline \vee$, i.e. $\min$, has been shown to be commutative in all cases.