Numbers which are Sum of Increasing Powers of Digits

Theorem
The following integers are the sum of the increasing powers of their digits taken in order begin as follows:
 * $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2 \, 646 \, 798, 12 \, 157 \, 692 \, 622 \, 039 \, 623 \, 539$

Proof
Single digit integers are trivial:
 * $d^1 = d$

for all $d \in \Z$.

Then we have:

Then:

and finally: