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Mirrors > Home > ILE Home > Th. List > zltnle | GIF version |
Description: 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zltnle | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9270 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
2 | zre 9270 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | lenlt 8046 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) | |
4 | 1, 2, 3 | syl2anr 290 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) |
5 | 4 | biimpd 144 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 → ¬ 𝐴 < 𝐵)) |
6 | 5 | con2d 625 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → ¬ 𝐵 ≤ 𝐴)) |
7 | ztri3or 9309 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | |
8 | ax-1 6 | . . . . 5 ⊢ (𝐴 < 𝐵 → (¬ 𝐵 ≤ 𝐴 → 𝐴 < 𝐵)) | |
9 | 8 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → (¬ 𝐵 ≤ 𝐴 → 𝐴 < 𝐵))) |
10 | eqcom 2189 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
11 | eqle 8062 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 = 𝐴) → 𝐵 ≤ 𝐴) | |
12 | 10, 11 | sylan2b 287 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐵 ≤ 𝐴) |
13 | 12 | ex 115 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐴 = 𝐵 → 𝐵 ≤ 𝐴)) |
14 | 13 | adantl 277 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → 𝐵 ≤ 𝐴)) |
15 | 1, 14 | sylan2 286 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 = 𝐵 → 𝐵 ≤ 𝐴)) |
16 | pm2.24 622 | . . . . 5 ⊢ (𝐵 ≤ 𝐴 → (¬ 𝐵 ≤ 𝐴 → 𝐴 < 𝐵)) | |
17 | 15, 16 | syl6 33 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 = 𝐵 → (¬ 𝐵 ≤ 𝐴 → 𝐴 < 𝐵))) |
18 | ltle 8058 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → 𝐵 ≤ 𝐴)) | |
19 | 1, 2, 18 | syl2anr 290 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 < 𝐴 → 𝐵 ≤ 𝐴)) |
20 | 19, 16 | syl6 33 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 < 𝐴 → (¬ 𝐵 ≤ 𝐴 → 𝐴 < 𝐵))) |
21 | 9, 17, 20 | 3jaod 1314 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → (¬ 𝐵 ≤ 𝐴 → 𝐴 < 𝐵))) |
22 | 7, 21 | mpd 13 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (¬ 𝐵 ≤ 𝐴 → 𝐴 < 𝐵)) |
23 | 6, 22 | impbid 129 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ w3o 978 = wceq 1363 ∈ wcel 2158 class class class wbr 4015 ℝcr 7823 < clt 8005 ≤ cle 8006 ℤcz 9266 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-n0 9190 df-z 9267 |
This theorem is referenced by: znnnlt1 9314 nn0n0n1ge2b 9345 eluzdc 9623 fzdcel 10053 fzn 10055 fzpreddisj 10084 fzp1disj 10093 fzneuz 10114 fznuz 10115 uznfz 10116 fzp1nel 10117 difelfznle 10148 fzodisj 10191 exfzdc 10253 modfzo0difsn 10408 fzfig 10443 iseqf1olemqk 10507 exp3val 10535 facdiv 10731 bcval5 10756 zfz1isolemiso 10832 2zsupmax 11247 2zinfmin 11264 summodclem3 11401 fprodntrivap 11605 alzdvds 11873 fzm1ndvds 11875 fzo0dvdseq 11876 n2dvds1 11930 dvdsbnd 11970 algcvgblem 12062 prmndvdsfaclt 12169 odzdvds 12258 pcprendvds 12303 pcdvdsb 12332 pc2dvds 12342 pcmpt 12354 pockthg 12368 prmunb 12373 1arith 12378 lgsdilem2 14708 uzdcinzz 14821 |
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