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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 9026 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
2 | zre 9026 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | lenlt 7808 | . . . . 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 598 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → ¬ 𝐵 ≤ 𝐴)) |
7 | ztri3or 9065 | . . 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 2119 | . . . . . . . . 9 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
11 | eqle 7823 | . . . . . . . . 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 | pm2.24 595 | . . . . 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 7819 | . . . . . 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 1267 | . . 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 946 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ℝcr 7587 < clt 7768 ≤ cle 7769 ℤcz 9022 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: znnnlt1 9070 nn0n0n1ge2b 9098 eluzdc 9372 fzdcel 9788 fzn 9790 fzpreddisj 9819 fzp1disj 9828 fzneuz 9849 fznuz 9850 uznfz 9851 fzp1nel 9852 difelfznle 9880 fzodisj 9923 exfzdc 9985 modfzo0difsn 10136 fzfig 10171 iseqf1olemqk 10235 exp3val 10263 facdiv 10452 bcval5 10477 zfz1isolemiso 10550 2zsupmax 10965 summodclem3 11117 alzdvds 11479 fzm1ndvds 11481 fzo0dvdseq 11482 n2dvds1 11536 dvdsbnd 11572 algcvgblem 11657 prmndvdsfaclt 11761 uzdcinzz 12932 |
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