Proving adsorption law

Question

Prove the absorption law

$A\cap(A\cup B)=A$

My Try:

Attempt 1:

$x\in A\cap(A\cup B)$

$x\in A$ and $x\in A\cup B\rightarrow x\in A$

Thus $A\cap(A\cup B)\subset A$

$x\in A\rightarrow x\in A$ and $x\in A\cup B\rightarrow x\in A\cap(A\cup B)$

Thus $A\subset A\cap(A\cup B)$

So, $A\cap(A\cup B)=A$

Attempt 2:

$x\in A\cap(A\cup B)\iff x\in A$ and $x\in A\cup B$

$\iff x\in A$ and $[x\in A$ or $x\in B]$

$\iff x\in A$

$A\cap(A\cup B)=A$

My Question: Which of my above attempts is most precise?

Answer 1

Instead of a point by point analysis
here is a set alegbra proof.

As U $\cap$ V $\subseteq$ U $\subseteq$ U $\cup$ V,
A $\cap$ (A $\cup$ B) $\subseteq$ A = A $\cap$ A $\subseteq$ A $\cap$ (A $\cup$ B).
Thus A = A $\cap$ (A $\cup$ B).

The dual theorem A = A $\cup$ (A $\cap$ B)
is mostly an application of DeMorgans rules or one
could adapt the above proof to prove the dual theorem.