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 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∉ dom 𝑊 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fzonel | ⊢ ¬ ( ♯ ‘ 𝑊 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) | |
| 2 | 1 | a1i | ⊢ ( 𝑊 ∈ Word 𝑉 → ¬ ( ♯ ‘ 𝑊 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
| 3 | wrddm | ⊢ ( 𝑊 ∈ Word 𝑉 → dom 𝑊 = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) | |
| 4 | 2 3 | neleqtrrd | ⊢ ( 𝑊 ∈ Word 𝑉 → ¬ ( ♯ ‘ 𝑊 ) ∈ dom 𝑊 ) |
| 5 | df-nel | ⊢ ( ( ♯ ‘ 𝑊 ) ∉ dom 𝑊 ↔ ¬ ( ♯ ‘ 𝑊 ) ∈ dom 𝑊 ) | |
| 6 | 4 5 | sylibr | ⊢ ( 𝑊 ∈ Word 𝑉 → ( ♯ ‘ 𝑊 ) ∉ dom 𝑊 ) |