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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 3759 | . 2 ⊢ (x = 1 → (1 ≤ x ↔ 1 ≤ 1)) | |
2 | breq2 3759 | . 2 ⊢ (x = y → (1 ≤ x ↔ 1 ≤ y)) | |
3 | breq2 3759 | . 2 ⊢ (x = (y + 1) → (1 ≤ x ↔ 1 ≤ (y + 1))) | |
4 | breq2 3759 | . 2 ⊢ (x = A → (1 ≤ x ↔ 1 ≤ A)) | |
5 | 1le1 7356 | . 2 ⊢ 1 ≤ 1 | |
6 | nnre 7702 | . . 3 ⊢ (y ∈ ℕ → y ∈ ℝ) | |
7 | recn 6812 | . . . . . 6 ⊢ (y ∈ ℝ → y ∈ ℂ) | |
8 | 7 | addid1d 6959 | . . . . 5 ⊢ (y ∈ ℝ → (y + 0) = y) |
9 | 8 | breq2d 3767 | . . . 4 ⊢ (y ∈ ℝ → (1 ≤ (y + 0) ↔ 1 ≤ y)) |
10 | 0lt1 6938 | . . . . . . . 8 ⊢ 0 < 1 | |
11 | 0re 6825 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
12 | 1re 6824 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
13 | axltadd 6886 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ y ∈ ℝ) → (0 < 1 → (y + 0) < (y + 1))) | |
14 | 11, 12, 13 | mp3an12 1221 | . . . . . . . 8 ⊢ (y ∈ ℝ → (0 < 1 → (y + 0) < (y + 1))) |
15 | 10, 14 | mpi 15 | . . . . . . 7 ⊢ (y ∈ ℝ → (y + 0) < (y + 1)) |
16 | readdcl 6805 | . . . . . . . . 9 ⊢ ((y ∈ ℝ ∧ 0 ∈ ℝ) → (y + 0) ∈ ℝ) | |
17 | 11, 16 | mpan2 401 | . . . . . . . 8 ⊢ (y ∈ ℝ → (y + 0) ∈ ℝ) |
18 | peano2re 6946 | . . . . . . . 8 ⊢ (y ∈ ℝ → (y + 1) ∈ ℝ) | |
19 | lttr 6889 | . . . . . . . . 9 ⊢ (((y + 0) ∈ ℝ ∧ (y + 1) ∈ ℝ ∧ 1 ∈ ℝ) → (((y + 0) < (y + 1) ∧ (y + 1) < 1) → (y + 0) < 1)) | |
20 | 12, 19 | mp3an3 1220 | . . . . . . . 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 560 | . . . . 5 ⊢ (y ∈ ℝ → (¬ (y + 0) < 1 → ¬ (y + 1) < 1)) |
24 | lenlt 6891 | . . . . . 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 6891 | . . . . . 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 7711 | 1 ⊢ (A ∈ ℕ → 1 ≤ A) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 class class class wbr 3755 (class class class)co 5455 ℝcr 6710 0cc0 6711 1c1 6712 + caddc 6714 < clt 6857 ≤ cle 6858 ℕcn 7695 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1re 6777 ax-addrcl 6780 ax-0id 6791 ax-rnegex 6792 ax-pre-ltirr 6795 ax-pre-lttrn 6797 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-inn 7696 |
This theorem is referenced by: nnle1eq1 7719 nngt0 7720 nnnlt1 7721 nnrecgt0 7732 nnge1d 7737 elnnnn0c 8003 elnnz1 8044 zltp1le 8074 elfz1b 8722 fzo1fzo0n0 8809 elfzom1elp1fzo 8828 fzo0sn0fzo1 8847 nnlesq 9009 |
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