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Mirrors > Home > ILE Home > Th. List > ztri3or0 | GIF version |
Description: Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
ztri3or0 | ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9241 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | 1 | simprbi 275 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
3 | idd 21 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 → 𝑁 = 0)) | |
4 | nngt0 8930 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
5 | 4 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 ∈ ℕ → 0 < 𝑁)) |
6 | nngt0 8930 | . . . . 5 ⊢ (-𝑁 ∈ ℕ → 0 < -𝑁) | |
7 | zre 9243 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
8 | 7 | lt0neg1d 8459 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ↔ 0 < -𝑁)) |
9 | 6, 8 | syl5ibr 156 | . . . 4 ⊢ (𝑁 ∈ ℤ → (-𝑁 ∈ ℕ → 𝑁 < 0)) |
10 | 3, 5, 9 | 3orim123d 1320 | . . 3 ⊢ (𝑁 ∈ ℤ → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → (𝑁 = 0 ∨ 0 < 𝑁 ∨ 𝑁 < 0))) |
11 | 2, 10 | mpd 13 | . 2 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ 0 < 𝑁 ∨ 𝑁 < 0)) |
12 | 3orrot 984 | . 2 ⊢ ((𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁) ↔ (𝑁 = 0 ∨ 0 < 𝑁 ∨ 𝑁 < 0)) | |
13 | 11, 12 | sylibr 134 | 1 ⊢ (𝑁 ∈ ℤ → (𝑁 < 0 ∨ 𝑁 = 0 ∨ 0 < 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ℝcr 7798 0cc0 7799 < clt 7979 -cneg 8116 ℕcn 8905 ℤcz 9239 |
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 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-distr 7903 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-cnre 7910 ax-pre-ltirr 7911 ax-pre-ltwlin 7912 ax-pre-lttrn 7913 ax-pre-ltadd 7915 |
This theorem depends on definitions: df-bi 117 df-3or 979 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-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-sub 8117 df-neg 8118 df-inn 8906 df-z 9240 |
This theorem is referenced by: ztri3or 9282 zdvdsdc 11800 divalglemex 11907 divalg 11909 bezoutlemmain 11979 mulgval 12872 mulgfng 12873 |
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