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Mirrors > Home > ILE Home > Th. List > wrdlndm | GIF version |
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 |
---|---|
wrdlndm | ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∉ dom 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzonel 10217 | . . . 4 ⊢ ¬ (♯‘𝑊) ∈ (0..^(♯‘𝑊)) | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → ¬ (♯‘𝑊) ∈ (0..^(♯‘𝑊))) |
3 | wrddm 10909 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → dom 𝑊 = (0..^(♯‘𝑊))) | |
4 | 2, 3 | neleqtrrd 2292 | . 2 ⊢ (𝑊 ∈ Word 𝑉 → ¬ (♯‘𝑊) ∈ dom 𝑊) |
5 | df-nel 2460 | . 2 ⊢ ((♯‘𝑊) ∉ dom 𝑊 ↔ ¬ (♯‘𝑊) ∈ dom 𝑊) | |
6 | 4, 5 | sylibr 134 | 1 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∉ dom 𝑊) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2164 ∉ wnel 2459 dom cdm 4655 ‘cfv 5246 (class class class)co 5910 0cc0 7862 ..^cfzo 10198 ♯chash 10833 Word cword 10901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-addcom 7962 ax-addass 7964 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-0id 7970 ax-rnegex 7971 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-recs 6349 df-frec 6435 df-1o 6460 df-er 6578 df-en 6786 df-dom 6787 df-fin 6788 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-inn 8973 df-n0 9231 df-z 9308 df-uz 9583 df-fz 10065 df-fzo 10199 df-ihash 10834 df-word 10902 |
This theorem is referenced by: (None) |
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