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Mirrors > Home > ILE Home > Th. List > nngt1ne1 | GIF version |
Description: A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.) |
Ref | Expression |
---|---|
nngt1ne1 | ⊢ (𝐴 ∈ ℕ → (1 < 𝐴 ↔ 𝐴 ≠ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7758 | . . 3 ⊢ 1 ∈ ℝ | |
2 | ltne 7842 | . . 3 ⊢ ((1 ∈ ℝ ∧ 1 < 𝐴) → 𝐴 ≠ 1) | |
3 | 1, 2 | mpan 420 | . 2 ⊢ (1 < 𝐴 → 𝐴 ≠ 1) |
4 | df-ne 2307 | . . 3 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1) | |
5 | nn1gt1 8747 | . . . 4 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴)) | |
6 | 5 | ord 713 | . . 3 ⊢ (𝐴 ∈ ℕ → (¬ 𝐴 = 1 → 1 < 𝐴)) |
7 | 4, 6 | syl5bi 151 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ≠ 1 → 1 < 𝐴)) |
8 | 3, 7 | impbid2 142 | 1 ⊢ (𝐴 ∈ ℕ → (1 < 𝐴 ↔ 𝐴 ≠ 1)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 class class class wbr 3924 ℝcr 7612 1c1 7614 < clt 7793 ℕcn 8713 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-inn 8714 |
This theorem is referenced by: prime 9143 eluz2b3 9391 ncoprmgcdne1b 11759 ncoprmgcdgt1b 11760 |
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