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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 11900 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnn1uz2 12326 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
3 | orc 863 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 = 1 ∨ 2 < 𝑁)) | |
4 | 3 | a1d 25 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
5 | 2z 12015 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | 5 | eluz1i 12252 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) |
7 | 2re 11712 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
9 | zre 11986 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
10 | 8, 9 | leloed 10783 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁))) |
11 | olc 864 | . . . . . . . . . . 11 ⊢ (2 < 𝑁 → (𝑁 = 1 ∨ 2 < 𝑁)) | |
12 | 11 | a1d 25 | . . . . . . . . . 10 ⊢ (2 < 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
13 | eleq1 2900 | . . . . . . . . . . . 12 ⊢ (𝑁 = 2 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) | |
14 | 13 | eqcoms 2829 | . . . . . . . . . . 11 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) |
15 | 2noddALTV 43878 | . . . . . . . . . . . 12 ⊢ 2 ∉ Odd | |
16 | df-nel 3124 | . . . . . . . . . . . . 13 ⊢ (2 ∉ Odd ↔ ¬ 2 ∈ Odd ) | |
17 | pm2.21 123 | . . . . . . . . . . . . 13 ⊢ (¬ 2 ∈ Odd → (2 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
18 | 16, 17 | sylbi 219 | . . . . . . . . . . . 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 | syl6bi 255 | . . . . . . . . . 10 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
21 | 12, 20 | jaoi 853 | . . . . . . . . 9 ⊢ ((2 < 𝑁 ∨ 2 = 𝑁) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
22 | 10, 21 | syl6bi 255 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁)))) |
23 | 22 | imp 409 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 2 ≤ 𝑁) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
24 | 6, 23 | sylbi 219 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
25 | 4, 24 | jaoi 853 | . . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
26 | 2, 25 | sylbi 219 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
27 | eleq1 2900 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
28 | 0noddALTV 43874 | . . . . . 6 ⊢ 0 ∉ Odd | |
29 | df-nel 3124 | . . . . . . 7 ⊢ (0 ∉ Odd ↔ ¬ 0 ∈ Odd ) | |
30 | pm2.21 123 | . . . . . . 7 ⊢ (¬ 0 ∈ Odd → (0 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
31 | 29, 30 | sylbi 219 | . . . . . 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 | syl6bi 255 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
34 | 26, 33 | jaoi 853 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
35 | 1, 34 | sylbi 219 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
36 | 35 | imp 409 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 class class class wbr 5066 ‘cfv 6355 ℝcr 10536 0cc0 10537 1c1 10538 < clt 10675 ≤ cle 10676 ℕcn 11638 2c2 11693 ℕ0cn0 11898 ℤcz 11982 ℤ≥cuz 12244 Odd codd 43810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-even 43811 df-odd 43812 |
This theorem is referenced by: (None) |
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