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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 4006 | . 2 ⊢ (𝑥 = 1 → (1 ≤ 𝑥 ↔ 1 ≤ 1)) | |
2 | breq2 4006 | . 2 ⊢ (𝑥 = 𝑦 → (1 ≤ 𝑥 ↔ 1 ≤ 𝑦)) | |
3 | breq2 4006 | . 2 ⊢ (𝑥 = (𝑦 + 1) → (1 ≤ 𝑥 ↔ 1 ≤ (𝑦 + 1))) | |
4 | breq2 4006 | . 2 ⊢ (𝑥 = 𝐴 → (1 ≤ 𝑥 ↔ 1 ≤ 𝐴)) | |
5 | 1le1 8524 | . 2 ⊢ 1 ≤ 1 | |
6 | nnre 8921 | . . 3 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
7 | recn 7940 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | 7 | addid1d 8101 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) = 𝑦) |
9 | 8 | breq2d 4014 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ 1 ≤ 𝑦)) |
10 | 0lt1 8079 | . . . . . . . 8 ⊢ 0 < 1 | |
11 | 0re 7953 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
12 | 1re 7952 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
13 | axltadd 8022 | . . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) | |
14 | 11, 12, 13 | mp3an12 1327 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (0 < 1 → (𝑦 + 0) < (𝑦 + 1))) |
15 | 10, 14 | mpi 15 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) < (𝑦 + 1)) |
16 | readdcl 7933 | . . . . . . . . 9 ⊢ ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → (𝑦 + 0) ∈ ℝ) | |
17 | 11, 16 | mpan2 425 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 0) ∈ ℝ) |
18 | peano2re 8088 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈ ℝ) | |
19 | lttr 8026 | . . . . . . . . 9 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ ∧ 1 ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) | |
20 | 12, 19 | mp3an3 1326 | . . . . . . . 8 ⊢ (((𝑦 + 0) ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
21 | 17, 18, 20 | syl2anc 411 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → (((𝑦 + 0) < (𝑦 + 1) ∧ (𝑦 + 1) < 1) → (𝑦 + 0) < 1)) |
22 | 15, 21 | mpand 429 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝑦 + 1) < 1 → (𝑦 + 0) < 1)) |
23 | 22 | con3d 631 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (¬ (𝑦 + 0) < 1 → ¬ (𝑦 + 1) < 1)) |
24 | lenlt 8028 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 0) ∈ ℝ) → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) | |
25 | 12, 17, 24 | sylancr 414 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) ↔ ¬ (𝑦 + 0) < 1)) |
26 | lenlt 8028 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) | |
27 | 12, 18, 26 | sylancr 414 | . . . . 5 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 1) ↔ ¬ (𝑦 + 1) < 1)) |
28 | 23, 25, 27 | 3imtr4d 203 | . . . 4 ⊢ (𝑦 ∈ ℝ → (1 ≤ (𝑦 + 0) → 1 ≤ (𝑦 + 1))) |
29 | 9, 28 | sylbird 170 | . . 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 8930 | 1 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 class class class wbr 4002 (class class class)co 5871 ℝcr 7806 0cc0 7807 1c1 7808 + caddc 7810 < clt 7987 ≤ cle 7988 ℕcn 8914 |
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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-1re 7901 ax-addrcl 7904 ax-0lt1 7913 ax-0id 7915 ax-rnegex 7916 ax-pre-ltirr 7919 ax-pre-lttrn 7921 ax-pre-ltadd 7923 |
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 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-xp 4631 df-cnv 4633 df-iota 5176 df-fv 5222 df-ov 5874 df-pnf 7989 df-mnf 7990 df-xr 7991 df-ltxr 7992 df-le 7993 df-inn 8915 |
This theorem is referenced by: nnle1eq1 8938 nngt0 8939 nnnlt1 8940 nnrecgt0 8952 nnge1d 8957 elnnnn0c 9216 elnnz1 9271 zltp1le 9302 nn0ledivnn 9762 elfz1b 10084 fzo1fzo0n0 10177 elfzom1elp1fzo 10196 fzo0sn0fzo1 10215 nnlesq 10617 faclbnd 10713 faclbnd3 10715 cvgratz 11532 coprmgcdb 12079 isprm3 12109 pw2dvds 12157 pockthg 12346 oddennn 12384 |
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