How can the complementation law in Table 1 be proven for \stackrel{=}{A} = A?

[Bashyboy]

Homework Statement

Prove the complementation law in Table 1 by showing
that $\stackrel{=}{A} = A$

Homework Equations

The Attempt at a Solution

Well, first I assumed that x is an element of A, so that $A = (x | x\in A)$

by taking the complement, I got $(x | \neg(x\in A) \rightarrow (x | x\notin A)$

then, taking the complement of the complement is where I get stuck:

$(x | \neg(x \in \overline{A})$

 

[Bashyboy]

I have another one:
Prove the domination laws in Table 1 by showing that
A ∪ U = U

A∪U = {x| x∈A∨x∈U} = {x| x∈ A ∨ T} = {x| T}=U

This is from the solution manual. I understand all but the last step. To me, the last step seems meaningless; how could you infer anything from it?

 

[Bashyboy]

In my original post, the arrow should actually be an equal sign.