If we define the set of all sets as: V={the x's such that (x is a set) & (x=x)}, then
V is a member of V means, V satisfies the predication ..(x is a set) & (x=x).

1. V is a set, is true by definition.

2. V=V, means, (x is a member of V <-> x is a member of V) for all x.
But, (all x)(Fx <-> Fx) is logically true, ie. V=V is tautologous.

therefore,

(V is a set) & (V=V), is true.
That is to say, V satisfies the predicate (x is a set) & (x=x).

The set of all sets is a member of itself.