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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 9003 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nnre 8751 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
3 | nngt0 8769 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
4 | 0re 7790 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | ltle 7875 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁)) | |
6 | 4, 5 | mpan 421 | . . . 4 ⊢ (𝑁 ∈ ℝ → (0 < 𝑁 → 0 ≤ 𝑁)) |
7 | 2, 3, 6 | sylc 62 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
8 | 0le0 8833 | . . . 4 ⊢ 0 ≤ 0 | |
9 | breq2 3941 | . . . 4 ⊢ (𝑁 = 0 → (0 ≤ 𝑁 ↔ 0 ≤ 0)) | |
10 | 8, 9 | mpbiri 167 | . . 3 ⊢ (𝑁 = 0 → 0 ≤ 𝑁) |
11 | 7, 10 | jaoi 706 | . 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 698 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ℝcr 7643 0cc0 7644 < clt 7824 ≤ cle 7825 ℕcn 8744 ℕ0cn0 9001 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-inn 8745 df-n0 9002 |
This theorem is referenced by: nn0nlt0 9027 nn0ge0i 9028 nn0le0eq0 9029 nn0p1gt0 9030 0mnnnnn0 9033 nn0addge1 9047 nn0addge2 9048 nn0ge0d 9057 elnn0z 9091 nn0lt10b 9155 nn0ge0div 9162 nn0pnfge0 9607 xnn0xadd0 9680 0elfz 9929 fz0fzelfz0 9935 fz0fzdiffz0 9938 fzctr 9941 difelfzle 9942 elfzodifsumelfzo 10009 fvinim0ffz 10049 subfzo0 10050 adddivflid 10096 modqmuladdnn0 10172 modfzo0difsn 10199 uzennn 10240 bernneq 10443 bernneq3 10445 faclbnd 10519 faclbnd6 10522 facubnd 10523 bcval5 10541 fihashneq0 10573 dvdseq 11582 evennn02n 11615 nn0ehalf 11636 nn0oddm1d2 11642 gcdn0gt0 11702 nn0gcdid0 11705 absmulgcd 11741 algcvgblem 11766 algcvga 11768 lcmgcdnn 11799 hashgcdlem 11939 znnen 11947 logbgcd1irr 13092 |
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