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Mirrors > Home > MPE Home > Th. List > wrdlenge2n0 | Structured version Visualization version GIF version |
Description: A word with length at least 2 is not empty. (Contributed by AV, 18-Jun-2018.) (Proof shortened by AV, 14-Oct-2018.) |
Ref | Expression |
---|---|
wrdlenge2n0 | ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 𝑊 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 11260 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → 1 ∈ ℝ) | |
2 | 2re 12338 | . . . . . 6 ⊢ 2 ∈ ℝ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → 2 ∈ ℝ) |
4 | lencl 14568 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ0) | |
5 | 4 | nn0red 12586 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℝ) |
6 | 1, 3, 5 | 3jca 1127 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ)) |
8 | simpr 484 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 2 ≤ (♯‘𝑊)) | |
9 | 1lt2 12435 | . . . 4 ⊢ 1 < 2 | |
10 | 8, 9 | jctil 519 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (1 < 2 ∧ 2 ≤ (♯‘𝑊))) |
11 | ltleletr 11352 | . . 3 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (♯‘𝑊) ∈ ℝ) → ((1 < 2 ∧ 2 ≤ (♯‘𝑊)) → 1 ≤ (♯‘𝑊))) | |
12 | 7, 10, 11 | sylc 65 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 1 ≤ (♯‘𝑊)) |
13 | wrdlenge1n0 14585 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (♯‘𝑊))) | |
14 | 13 | adantr 480 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 ≠ ∅ ↔ 1 ≤ (♯‘𝑊))) |
15 | 12, 14 | mpbird 257 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 𝑊 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 ℝcr 11152 1c1 11154 < clt 11293 ≤ cle 11294 2c2 12319 ♯chash 14366 Word cword 14549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 |
This theorem is referenced by: pfxtrcfv0 14729 clwlkclwwlk 30031 pfx1s2 32908 pfxlsw2ccat 32920 |
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