Integer Addition Identity is Zero

Theorem
The identity of integer addition is $0$:
 * $\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$

Proof
Let us define $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxtimes$ as in the formal definition of integers.

That is, $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxtimes$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxtimes$.

$\boxtimes$ is the congruence relation defined on $\N \times \N$ by $\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$.

In order to streamline the notation, we will use $\left[\!\left[{a, b}\right]\!\right]$ to mean $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxtimes$, as suggested.

From the method of construction, $\left[\!\left[{c, c}\right]\!\right]$, where $c$ is any element of the natural numbers $\N$, is the isomorphic copy of $0 \in \N$.

So, we need to show that $\forall a, b, c \in \N: \left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{c, c}\right]\!\right] = \left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{c, c}\right]\!\right] + \left[\!\left[{a, b}\right]\!\right]$.

Thus:

So $\left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{c, c}\right]\!\right] = \left[\!\left[{a, b}\right]\!\right]$.

The identity $\left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{c, c}\right]\!\right] + \left[\!\left[{a, b}\right]\!\right]$ is demonstrated similarly.