Vitali Set Existence Theorem/Lemma

Theorem
For all real numbers in the closed unit interval $\mathbb I = \closedint 0 1$, define the relation $\sim$ such that:
 * $\forall x, y \in \mathbb I: x \sim y \iff x - y \in \Q$

where $\Q$ is the set of rational numbers.

That is, $x \sim y$ their difference is rational.

Then $\sim$ is an equivalence relation.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity

 * $\forall x \in \mathbb I: x - x = 0 \in \Q$

Thus $\sim$ is reflexive.

Symmetry
By Inverse for Rational Addition:
 * $\forall x, y \in \mathbb I: x - y \in \Q \implies y - x = -\paren {x - y} \in \Q$

Thus $\sim$ is symmetric.

Transitivity
Let $x, y, z \in \mathbb I$.

Suppose $x - y \in \Q$ and $y - z \in \Q$.

Then by Rational Addition is Closed:
 * $x - z = \paren {x - y} + \paren {y - z} \in \Q$

Thus $\sim$ is transitive.

Thus we have shown that $\sim$ is an equivalence relation.