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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 7955 | . . 3 ⊢ 1 ∈ ℝ | |
2 | ltne 8041 | . . 3 ⊢ ((1 ∈ ℝ ∧ 1 < 𝐴) → 𝐴 ≠ 1) | |
3 | 1, 2 | mpan 424 | . 2 ⊢ (1 < 𝐴 → 𝐴 ≠ 1) |
4 | df-ne 2348 | . . 3 ⊢ (𝐴 ≠ 1 ↔ ¬ 𝐴 = 1) | |
5 | nn1gt1 8952 | . . . 4 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ 1 < 𝐴)) | |
6 | 5 | ord 724 | . . 3 ⊢ (𝐴 ∈ ℕ → (¬ 𝐴 = 1 → 1 < 𝐴)) |
7 | 4, 6 | biimtrid 152 | . 2 ⊢ (𝐴 ∈ ℕ → (𝐴 ≠ 1 → 1 < 𝐴)) |
8 | 3, 7 | impbid2 143 | 1 ⊢ (𝐴 ∈ ℕ → (1 < 𝐴 ↔ 𝐴 ≠ 1)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4003 ℝcr 7809 1c1 7811 < clt 7991 ℕcn 8918 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 4004 df-opab 4065 df-xp 4632 df-cnv 4634 df-iota 5178 df-fv 5224 df-ov 5877 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-inn 8919 |
This theorem is referenced by: prime 9351 eluz2b3 9603 ncoprmgcdne1b 12088 ncoprmgcdgt1b 12089 |
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