Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 12181 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnn1uz2 12610 | . . . . 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 12298 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | 5 | eluz1i 12535 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) |
7 | 2re 11993 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
9 | zre 12269 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
10 | 8, 9 | leloed 11064 | . . . . . . . . 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 2824 | . . . . . . . . . . . 12 ⊢ (𝑁 = 2 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) | |
14 | 13 | eqcoms 2745 | . . . . . . . . . . 11 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) |
15 | 2noddALTV 45075 | . . . . . . . . . . . 12 ⊢ 2 ∉ Odd | |
16 | df-nel 3048 | . . . . . . . . . . . . 13 ⊢ (2 ∉ Odd ↔ ¬ 2 ∈ Odd ) | |
17 | pm2.21 123 | . . . . . . . . . . . . 13 ⊢ (¬ 2 ∈ Odd → (2 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
18 | 16, 17 | sylbi 216 | . . . . . . . . . . . 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 252 | . . . . . . . . . 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 252 | . . . . . . . 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 216 | . . . . . 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 216 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
27 | eleq1 2824 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
28 | 0noddALTV 45071 | . . . . . 6 ⊢ 0 ∉ Odd | |
29 | df-nel 3048 | . . . . . . 7 ⊢ (0 ∉ Odd ↔ ¬ 0 ∈ Odd ) | |
30 | pm2.21 123 | . . . . . . 7 ⊢ (¬ 0 ∈ Odd → (0 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
31 | 29, 30 | sylbi 216 | . . . . . 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 252 | . . . 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 216 | . 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 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2107 ∉ wnel 3047 class class class wbr 5075 ‘cfv 6423 ℝcr 10817 0cc0 10818 1c1 10819 < clt 10956 ≤ cle 10957 ℕcn 11919 2c2 11974 ℕ0cn0 12179 ℤcz 12265 ℤ≥cuz 12527 Odd codd 45007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-om 7693 df-2nd 7810 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-er 8461 df-en 8697 df-dom 8698 df-sdom 8699 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-div 11579 df-nn 11920 df-2 11982 df-n0 12180 df-z 12266 df-uz 12528 df-even 45008 df-odd 45009 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |