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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 11902 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnn1uz2 12328 | . . . . 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 12017 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | 5 | eluz1i 12254 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) |
7 | 2re 11714 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
9 | zre 11988 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
10 | 8, 9 | leloed 10785 | . . . . . . . . 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 2902 | . . . . . . . . . . . 12 ⊢ (𝑁 = 2 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) | |
14 | 13 | eqcoms 2831 | . . . . . . . . . . 11 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) |
15 | 2noddALTV 43865 | . . . . . . . . . . . 12 ⊢ 2 ∉ Odd | |
16 | df-nel 3126 | . . . . . . . . . . . . 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 2902 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
28 | 0noddALTV 43861 | . . . . . 6 ⊢ 0 ∉ Odd | |
29 | df-nel 3126 | . . . . . . 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 3125 class class class wbr 5068 ‘cfv 6357 ℝcr 10538 0cc0 10539 1c1 10540 < clt 10677 ≤ cle 10678 ℕcn 11640 2c2 11695 ℕ0cn0 11900 ℤcz 11984 ℤ≥cuz 12246 Odd codd 43797 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-even 43798 df-odd 43799 |
This theorem is referenced by: (None) |
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