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Mirrors > Home > ILE Home > Th. List > nn0ge0 | GIF version |
Description: A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn0ge0 | ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8947 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nnre 8695 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
3 | nngt0 8713 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
4 | 0re 7734 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | ltle 7819 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁)) | |
6 | 4, 5 | mpan 420 | . . . 4 ⊢ (𝑁 ∈ ℝ → (0 < 𝑁 → 0 ≤ 𝑁)) |
7 | 2, 3, 6 | sylc 62 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
8 | 0le0 8777 | . . . 4 ⊢ 0 ≤ 0 | |
9 | breq2 3903 | . . . 4 ⊢ (𝑁 = 0 → (0 ≤ 𝑁 ↔ 0 ≤ 0)) | |
10 | 8, 9 | mpbiri 167 | . . 3 ⊢ (𝑁 = 0 → 0 ≤ 𝑁) |
11 | 7, 10 | jaoi 690 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → 0 ≤ 𝑁) |
12 | 1, 11 | sylbi 120 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 682 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ℝcr 7587 0cc0 7588 < clt 7768 ≤ cle 7769 ℕcn 8688 ℕ0cn0 8945 |
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-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 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-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-rab 2402 df-v 2662 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-xp 4515 df-cnv 4517 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-inn 8689 df-n0 8946 |
This theorem is referenced by: nn0nlt0 8971 nn0ge0i 8972 nn0le0eq0 8973 nn0p1gt0 8974 0mnnnnn0 8977 nn0addge1 8991 nn0addge2 8992 nn0ge0d 9001 elnn0z 9035 nn0lt10b 9099 nn0ge0div 9106 nn0pnfge0 9545 xnn0xadd0 9618 0elfz 9866 fz0fzelfz0 9872 fz0fzdiffz0 9875 fzctr 9878 difelfzle 9879 elfzodifsumelfzo 9946 fvinim0ffz 9986 subfzo0 9987 adddivflid 10033 modqmuladdnn0 10109 modfzo0difsn 10136 uzennn 10177 bernneq 10380 bernneq3 10382 faclbnd 10455 faclbnd6 10458 facubnd 10459 bcval5 10477 fihashneq0 10509 dvdseq 11473 evennn02n 11506 nn0ehalf 11527 nn0oddm1d2 11533 gcdn0gt0 11593 nn0gcdid0 11596 absmulgcd 11632 algcvgblem 11657 algcvga 11659 lcmgcdnn 11690 hashgcdlem 11830 znnen 11838 |
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