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 $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$ as in the formal definition of integers.

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

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

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