Description: The length of a word is not in the domain of the word (regarded as a function). (Contributed by AV, 3-Mar-2021) (Proof shortened by JJ, 18-Nov-2022)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wrdlndm | |- ( W e. Word V -> ( # ` W ) e/ dom W ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fzonel | |- -. ( # ` W ) e. ( 0 ..^ ( # ` W ) ) |
|
| 2 | 1 | a1i | |- ( W e. Word V -> -. ( # ` W ) e. ( 0 ..^ ( # ` W ) ) ) |
| 3 | wrddm | |- ( W e. Word V -> dom W = ( 0 ..^ ( # ` W ) ) ) |
|
| 4 | 2 3 | neleqtrrd | |- ( W e. Word V -> -. ( # ` W ) e. dom W ) |
| 5 | df-nel | |- ( ( # ` W ) e/ dom W <-> -. ( # ` W ) e. dom W ) |
|
| 6 | 4 5 | sylibr | |- ( W e. Word V -> ( # ` W ) e/ dom W ) |