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Mirrors > Home > ILE Home > Th. List > nnnlt1 | GIF version |
Description: A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnnlt1 | ⊢ (𝐴 ∈ ℕ → ¬ 𝐴 < 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1 8871 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
2 | 1re 7889 | . . 3 ⊢ 1 ∈ ℝ | |
3 | nnre 8855 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
4 | lenlt 7965 | . . 3 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (1 ≤ 𝐴 ↔ ¬ 𝐴 < 1)) | |
5 | 2, 3, 4 | sylancr 411 | . 2 ⊢ (𝐴 ∈ ℕ → (1 ≤ 𝐴 ↔ ¬ 𝐴 < 1)) |
6 | 1, 5 | mpbid 146 | 1 ⊢ (𝐴 ∈ ℕ → ¬ 𝐴 < 1) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 2135 class class class wbr 3976 ℝcr 7743 1c1 7745 < clt 7924 ≤ cle 7925 ℕcn 8848 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-iota 5147 df-fv 5190 df-ov 5839 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-inn 8849 |
This theorem is referenced by: 0nnn 8875 nnsub 8887 indstr 9522 indstr2 9538 sqrt2irr 12071 |
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