If, in a lattice, join and meet of $x$ and $y$ coincide, then $x=y$

Question

Prove that in any lattice $(L,\leq)$ $$\forall x,y \in L ((x \wedge y = x\vee y) \Rightarrow (x = y)).$$

Could anyone give me an advice?

Answer 1

Call $a=x\vee y$. $$\begin{cases}a=(x\wedge y)\le x\le (x\vee y)=a\\a=(x\wedge y)\le y\le (x\vee y)=a\end{cases}$$ So $x=a$ and $y=a$.

Answer 2

$$x=x\wedge(x\vee y)=x\wedge(x\wedge y)=x\wedge y=(x\wedge y)\wedge y=(x\vee y)\wedge y=y$$