Although this is not the “detailed proof” you seek, you might still find Chaitin’s own reasoning interesting, as articulated in his Scientific American article “The Limits of Reason” (PDF):

Why Is Omega Incompressible?

I wish to demonstrate that omega is incompressible—that one cannot use a program substantially shorter than N bits long to compute the first N bits of omega. The demonstration will involve a careful combination of facts about omega and the Turing halting problem that it is so intimately related to. Specifically, I will use the fact that the halting problem for programs up to length N bits cannot be solved by a program that is itself shorter than N bits (see www.sciam.com/ontheweb)

My strategy for demonstrating that omega is incompressible is to show that having the first N bits of omega would tell me how to solve the Turing halting problem for programs up to length N bits. It follows from that conclusion that no program shorter than N bits can compute the first N bits of omega. (If such a program existed, I could use it to compute the first N bits of omega and then use those bits to solve Turing’s problem up to N bits—a task that is impossible for such a short program.)

Now let us see how knowing N bits of omega would enable me to solve the halting problem—to determine which programs halt—for all programs up to N bits in size. Do this by performing a computation in stages. Use the integer K to label which stage we are at: K = 1, 2, 3, …

At stage K, run every program up to K bits in size for K seconds. Then compute a halting probability, which we will call omega_K, based on all the programs that halt by stage K. Omega_K will be less than omega because it is based on only a subset of all the programs that halt eventually, whereas omega is based on all such programs.

As K increases, the value of omega_K will get closer and closer to the actual value of omega. As it gets closer to omega’s actual value, more and more of omega_K’s first bits will be correct—that is, the same as the corresponding bits of omega.

And as soon as the first N bits are correct, you know that you have encountered every program up to N bits in size that will ever halt. (If there were another such N-bit program, at some later-stage K that program would halt, which would increase the value of omega_K to be greater than omega, which is impossible.)

So we can use the first N bits of omega to solve the halting problem for all programs up to N bits in size. Now suppose we could compute the first N bits of omega with a program substantially shorter than N bits long. We could then combine that program with the one for carrying out the omega_K algorithm, to produce a program shorter than N bits that solves the Turing halting problem up to programs of length N bits.

But, as stated up front, we know that no such program exists. Consequently, the first N bits of omega must require a program that is almost N bits long to compute them. That is good enough to call omega incompressible or irreducible. (A compression from N bits to almost N bits is not significant for large N.)