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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 9181 | . 2 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | nnre 8929 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
3 | nngt0 8947 | . . . 4 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
4 | 0re 7960 | . . . . 5 ⊢ 0 ∈ ℝ | |
5 | ltle 8048 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑁 → 0 ≤ 𝑁)) | |
6 | 4, 5 | mpan 424 | . . . 4 ⊢ (𝑁 ∈ ℝ → (0 < 𝑁 → 0 ≤ 𝑁)) |
7 | 2, 3, 6 | sylc 62 | . . 3 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
8 | 0le0 9011 | . . . 4 ⊢ 0 ≤ 0 | |
9 | breq2 4009 | . . . 4 ⊢ (𝑁 = 0 → (0 ≤ 𝑁 ↔ 0 ≤ 0)) | |
10 | 8, 9 | mpbiri 168 | . . 3 ⊢ (𝑁 = 0 → 0 ≤ 𝑁) |
11 | 7, 10 | jaoi 716 | . 2 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → 0 ≤ 𝑁) |
12 | 1, 11 | sylbi 121 | 1 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 = wceq 1353 ∈ wcel 2148 class class class wbr 4005 ℝcr 7813 0cc0 7814 < clt 7995 ≤ cle 7996 ℕcn 8922 ℕ0cn0 9179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-iota 5180 df-fv 5226 df-ov 5881 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-inn 8923 df-n0 9180 |
This theorem is referenced by: nn0nlt0 9205 nn0ge0i 9206 nn0le0eq0 9207 nn0p1gt0 9208 0mnnnnn0 9211 nn0addge1 9225 nn0addge2 9226 nn0ge0d 9235 elnn0z 9269 nn0negleid 9324 nn0lt10b 9336 nn0ge0div 9343 nn0pnfge0 9794 xnn0xadd0 9870 0elfz 10121 fz0fzelfz0 10130 fz0fzdiffz0 10133 fzctr 10136 difelfzle 10137 elfzodifsumelfzo 10204 fvinim0ffz 10244 subfzo0 10245 adddivflid 10295 modqmuladdnn0 10371 modfzo0difsn 10398 uzennn 10439 bernneq 10644 bernneq3 10646 faclbnd 10724 faclbnd6 10727 facubnd 10728 bcval5 10746 fihashneq0 10777 dvdseq 11857 evennn02n 11890 nn0ehalf 11911 nn0oddm1d2 11917 gcdn0gt0 11982 nn0gcdid0 11985 absmulgcd 12021 algcvgblem 12052 algcvga 12054 lcmgcdnn 12085 hashgcdlem 12241 odzdvds 12248 pcfaclem 12350 znnen 12402 logbgcd1irr 14573 lgsdinn0 14637 |
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