Cartesian Product is not Associative/Proof 2

Formal Proof
Assign to every set $X$ the following number $\map n X \in \N$:


 * $\map n X = \begin{cases}

0 & : X = \O \\ \displaystyle 1 + \max_{Y \mathop \in X} \map n Y & : \text{ otherwise} \end{cases}$

From the Axiom of Foundation:
 * $\forall X \in \N: \map n X < \infty$

Now let $a \in A$ be such that:
 * $\displaystyle \map n a = \min_{b \mathop \in A} \map n b$

Suppose that:
 * $\exists a' \in A, b \in B: a = \tuple {a', b}$

That is, that $a$ equals the ordered pair of $a'$ and $b'$.

Then it follows that:

That is:
 * $\map n {a'} < \map n a$

contradicting the assumed minimality of the latter.

Therefore:
 * $a \notin A \times B$

and hence:
 * $A \nsubseteq A \times B$

It follows from Equality of Cartesian Products that:
 * $A \times \paren {B \times C} \ne \paren {A \times B} \times C$