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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 9187 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
2 | zre 9187 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | lenlt 7966 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) | |
4 | 1, 2, 3 | syl2anr 288 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵)) |
5 | 4 | biimpd 143 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 → ¬ 𝐴 < 𝐵)) |
6 | 5 | con2d 614 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → ¬ 𝐵 ≤ 𝐴)) |
7 | ztri3or 9226 | . . 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 2166 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
11 | eqle 7982 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 = 𝐴) → 𝐵 ≤ 𝐴) | |
12 | 10, 11 | sylan2b 285 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 = 𝐵) → 𝐵 ≤ 𝐴) |
13 | 12 | ex 114 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → (𝐴 = 𝐵 → 𝐵 ≤ 𝐴)) |
14 | 13 | adantl 275 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 → 𝐵 ≤ 𝐴)) |
15 | 1, 14 | sylan2 284 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 = 𝐵 → 𝐵 ≤ 𝐴)) |
16 | pm2.24 611 | . . . . 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 7978 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → 𝐵 ≤ 𝐴)) | |
19 | 1, 2, 18 | syl2anr 288 | . . . . 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 1293 | . . 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 128 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 966 = wceq 1342 ∈ wcel 2135 class class class wbr 3977 ℝcr 7744 < clt 7925 ≤ cle 7926 ℤcz 9183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-iota 5148 df-fun 5185 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-n0 9107 df-z 9184 |
This theorem is referenced by: znnnlt1 9231 nn0n0n1ge2b 9262 eluzdc 9540 fzdcel 9966 fzn 9968 fzpreddisj 9997 fzp1disj 10006 fzneuz 10027 fznuz 10028 uznfz 10029 fzp1nel 10030 difelfznle 10061 fzodisj 10104 exfzdc 10166 modfzo0difsn 10321 fzfig 10356 iseqf1olemqk 10420 exp3val 10448 facdiv 10641 bcval5 10666 zfz1isolemiso 10742 2zsupmax 11157 2zinfmin 11174 summodclem3 11311 fprodntrivap 11515 alzdvds 11781 fzm1ndvds 11783 fzo0dvdseq 11784 n2dvds1 11838 dvdsbnd 11878 algcvgblem 11970 prmndvdsfaclt 12077 odzdvds 12166 pcprendvds 12211 pcdvdsb 12240 pc2dvds 12250 pcmpt 12262 pockthg 12276 prmunb 12281 1arith 12286 uzdcinzz 13541 |
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