Proving that idempotence follows from other lattice axioms

Question

I am supposed to prove that $x\wedge x = x$ and $x\vee x = x$ follow from the other lattice axioms (associativity, commutativity and absorption).

So far I have $$x = x\vee(x\wedge x) = x\vee(x\wedge(x\wedge(x\vee x))) = x\vee ((x\wedge x)\wedge(x\vee x))$$

My argumentation then was that since $x=x\vee(x\wedge x)$ that either $x\vee x$ or $x\wedge x$ has to be equal to x, after which the other idempotent law would follow. The problem is, that can't be right - $x\vee(y\wedge z)= x$ can be true even though neither y nor z is equal to x (trivial example, $x\neq\emptyset,y=\emptyset,z=\emptyset, x\cup(y\cap z) = x$. I can't really think of anything else either though...

Answer 1

If $x$ can be written as $x=x\land y$ for any $y$, then we have $$x=x\lor (x\land y) =x\lor x.$$ Now choose $y:=(x\lor x)$.