A Turing-recognizable language consisting of descriptions of Turing machines

Problem:

Let $A$ be a Turing-recognizable language consisting of descriptions of Turing machines, $\{(M_1 ), (M_2), \cdots \}$, where every $M_i$ is a decider. Prove that some decidable language $D$ is not decided by any decider $M_i$ whose description appears in $A$.
(Hint: You may find it helpful to consider an enumerator for $A$.)

Short answer: 

We can adopt the diagonalization argument to produce a decidable language that is not accepted by any of the $M_i$.

Proof:

Consider the language $D$ decided by the following Turing machine

Reject any input that is not a binary number.
Enumerate the ith Turing machine, where $i$ is the input string.
If $M_i$ accept the input string, reject it, otherwise accept it.

Apparently this is a Turing machine, and it is not accepted by any of the $M_i$ (there is always a string $i$ where $M_i$ does not agree with $D$)