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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0o1gt2ALTV | Structured version Visualization version GIF version | ||
| Description: An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.) |
| Ref | Expression |
|---|---|
| nn0o1gt2ALTV | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elnn0 12528 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
| 2 | elnn1uz2 12967 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
| 3 | orc 868 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 = 1 ∨ 2 < 𝑁)) | |
| 4 | 3 | a1d 25 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 5 | 2z 12649 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
| 6 | 5 | eluz1i 12886 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) |
| 7 | 2re 12340 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
| 8 | 7 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
| 9 | zre 12617 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
| 10 | 8, 9 | leloed 11404 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁))) |
| 11 | olc 869 | . . . . . . . . . . 11 ⊢ (2 < 𝑁 → (𝑁 = 1 ∨ 2 < 𝑁)) | |
| 12 | 11 | a1d 25 | . . . . . . . . . 10 ⊢ (2 < 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 13 | eleq1 2829 | . . . . . . . . . . . 12 ⊢ (𝑁 = 2 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) | |
| 14 | 13 | eqcoms 2745 | . . . . . . . . . . 11 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) |
| 15 | 2noddALTV 47680 | . . . . . . . . . . . 12 ⊢ 2 ∉ Odd | |
| 16 | df-nel 3047 | . . . . . . . . . . . . 13 ⊢ (2 ∉ Odd ↔ ¬ 2 ∈ Odd ) | |
| 17 | pm2.21 123 | . . . . . . . . . . . . 13 ⊢ (¬ 2 ∈ Odd → (2 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
| 18 | 16, 17 | sylbi 217 | . . . . . . . . . . . 12 ⊢ (2 ∉ Odd → (2 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 19 | 15, 18 | ax-mp 5 | . . . . . . . . . . 11 ⊢ (2 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁)) |
| 20 | 14, 19 | biimtrdi 253 | . . . . . . . . . 10 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 21 | 12, 20 | jaoi 858 | . . . . . . . . 9 ⊢ ((2 < 𝑁 ∨ 2 = 𝑁) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 22 | 10, 21 | biimtrdi 253 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁)))) |
| 23 | 22 | imp 406 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 2 ≤ 𝑁) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 24 | 6, 23 | sylbi 217 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 25 | 4, 24 | jaoi 858 | . . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 26 | 2, 25 | sylbi 217 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 27 | eleq1 2829 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
| 28 | 0noddALTV 47676 | . . . . . 6 ⊢ 0 ∉ Odd | |
| 29 | df-nel 3047 | . . . . . . 7 ⊢ (0 ∉ Odd ↔ ¬ 0 ∈ Odd ) | |
| 30 | pm2.21 123 | . . . . . . 7 ⊢ (¬ 0 ∈ Odd → (0 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
| 31 | 29, 30 | sylbi 217 | . . . . . 6 ⊢ (0 ∉ Odd → (0 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 32 | 28, 31 | ax-mp 5 | . . . . 5 ⊢ (0 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁)) |
| 33 | 27, 32 | biimtrdi 253 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 34 | 26, 33 | jaoi 858 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 35 | 1, 34 | sylbi 217 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
| 36 | 35 | imp 406 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 class class class wbr 5143 ‘cfv 6561 ℝcr 11154 0cc0 11155 1c1 11156 < clt 11295 ≤ cle 11296 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ℤcz 12613 ℤ≥cuz 12878 Odd codd 47612 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-even 47613 df-odd 47614 |
| This theorem is referenced by: (None) |
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