Proof that $ x \leq y$ and $y \leq x \Rightarrow x =$ $y$

Question

I need to prove that if $ x \leq y$ and $y \leq x \Rightarrow x =$ $y$.

I tried to prove this by contradiction.

Proof
Suppose that $ x \leq y$ and $y \leq x \Rightarrow x \neq y$. If $x \neq y$, we know out of the properties of inequalities that $x<y$ or $y<x$, which is the contradiction. Therefore $x \leq y$ and $y \leq x \Rightarrow x =$ $y$.

Is this proof correct and what other ways are there to prove this?

Answer 1

$y\ge x$ and $x \ge y$ gives us $y\ge x \ge y$ because $y=y$, $\;$ $y=x$ must also hold

Answer 2

For $x\ge y$ : Let$$x = y+a \quad\quad\quad \text{(1.)}$$ For $y\ge x$ : Let$$y = x+b \quad\quad\quad \text{(2.)}$$

Substituting $(1.)$ in $(2.)$

$$y = y+a+b$$ yielding $a=-b$. Putting this in $(1.) $ and $(2.)$ $$x=y-b$$

And $$y=x+b$$

Adding this results in $$2x=2y \implies \boxed{\color{blue}{x=y}}$$