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Mirrors > Home > ILE Home > Th. List > nnge1 | GIF version |
Description: A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
nnge1 | ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3768 | . 2 ⊢ (𝑥 = 1 → (1 ≤ 𝑥 ↔ 1 ≤ 1)) | |
2 | breq2 3768 | . 2 ⊢ (𝑥 = 𝑦 → (1 ≤ 𝑥 ↔ 1 ≤ 𝑦)) | |
3 | breq2 3768 | . 2 ⊢ (𝑥 = (𝑦 + 1) → (1 ≤ 𝑥 ↔ 1 ≤ (𝑦 + 1))) | |
4 | breq2 3768 | . 2 ⊢ (𝑥 = 𝐴 → (1 ≤ 𝑥 ↔ 1 ≤ 𝐴)) | |
5 | 1le1 7563 | . 2 ⊢ 1 ≤ 1 | |
6 | nnre 7921 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
7 | recn 7014 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | 7 | addid1d 7162 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) = 𝑦) |
9 | 8 | breq2d 3776 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ 1 ≤ 𝑦)) |
10 | 0lt1 7141 | . . . . . . . 8 ⊢ 0 < 1 | |
11 | 0re 7027 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
12 | 1re 7026 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
13 | axltadd 7089 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) | |
14 | 11, 12, 13 | mp3an12 1222 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) |
15 | 10, 14 | mpi 15 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) < (𝑦 + 1)) |
16 | readdcl 7007 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑦 + 0) ∈ ℝ) | |
17 | 11, 16 | mpan2 401 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) ∈ ℝ) |
18 | peano2re 7149 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ) | |
19 | lttr 7092 | . . . . . . . . 9 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ ∧ 1 ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) | |
20 | 12, 19 | mp3an3 1221 | . . . . . . . 8 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
21 | 17, 18, 20 | syl2anc 391 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
22 | 15, 21 | mpand 405 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝑦 + 1) < 1 → (𝑦 + 0) < 1)) |
23 | 22 | con3d 561 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (¬ (𝑦 + 0) < 1 → ¬ (𝑦 + 1) < 1)) |
24 | lenlt 7094 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 0) ∈ ℝ) → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) | |
25 | 12, 17, 24 | sylancr 393 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) |
26 | lenlt 7094 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) | |
27 | 12, 18, 26 | sylancr 393 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) |
28 | 23, 25, 27 | 3imtr4d 192 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) → 1 ≤ (𝑦 + 1))) |
29 | 9, 28 | sylbird 159 | . . 3 ⊢ (𝑦 ∈ ℝ → (1 ≤ 𝑦 → 1 ≤ (𝑦 + 1))) |
30 | 6, 29 | syl 14 | . 2 ⊢ (𝑦 ∈ ℕ → (1 ≤ 𝑦 → 1 ≤ (𝑦 + 1))) |
31 | 1, 2, 3, 4, 5, 30 | nnind 7930 | 1 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 ℝcr 6888 0cc0 6889 1c1 6890 + caddc 6892 < clt 7060 ≤ cle 7061 ℕcn 7914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1re 6978 ax-addrcl 6981 ax-0id 6992 ax-rnegex 6993 ax-pre-ltirr 6996 ax-pre-lttrn 6998 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-inn 7915 |
This theorem is referenced by: nnle1eq1 7938 nngt0 7939 nnnlt1 7940 nnrecgt0 7951 nnge1d 7956 elnnnn0c 8227 elnnz1 8268 zltp1le 8298 elfz1b 8952 fzo1fzo0n0 9039 elfzom1elp1fzo 9058 fzo0sn0fzo1 9077 nnlesq 9356 |
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