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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 8901 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
2 | 1re 7919 | . . 3 ⊢ 1 ∈ ℝ | |
3 | nnre 8885 | . . 3 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
4 | lenlt 7995 | . . 3 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (1 ≤ 𝐴 ↔ ¬ 𝐴 < 1)) | |
5 | 2, 3, 4 | sylancr 412 | . 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 2141 class class class wbr 3989 ℝcr 7773 1c1 7775 < clt 7954 ≤ cle 7955 ℕcn 8878 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-pre-ltirr 7886 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-iota 5160 df-fv 5206 df-ov 5856 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-inn 8879 |
This theorem is referenced by: 0nnn 8905 nnsub 8917 indstr 9552 indstr2 9568 sqrt2irr 12116 |
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