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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 11171 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | elnn1uz2 11641 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) | |
3 | orc 399 | . . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 = 1 ∨ 2 < 𝑁)) | |
4 | 3 | a1d 25 | . . . . . 6 ⊢ (𝑁 = 1 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
5 | 2z 11286 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | 5 | eluz1i 11571 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) |
7 | 2re 10967 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℝ) |
9 | zre 11258 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
10 | 8, 9 | leloed 10059 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 ↔ (2 < 𝑁 ∨ 2 = 𝑁))) |
11 | olc 398 | . . . . . . . . . . 11 ⊢ (2 < 𝑁 → (𝑁 = 1 ∨ 2 < 𝑁)) | |
12 | 11 | a1d 25 | . . . . . . . . . 10 ⊢ (2 < 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
13 | eleq1 2676 | . . . . . . . . . . . 12 ⊢ (𝑁 = 2 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) | |
14 | 13 | eqcoms 2618 | . . . . . . . . . . 11 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd ↔ 2 ∈ Odd )) |
15 | 2noddALTV 40142 | . . . . . . . . . . . 12 ⊢ 2 ∉ Odd | |
16 | df-nel 2783 | . . . . . . . . . . . . 13 ⊢ (2 ∉ Odd ↔ ¬ 2 ∈ Odd ) | |
17 | pm2.21 119 | . . . . . . . . . . . . 13 ⊢ (¬ 2 ∈ Odd → (2 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
18 | 16, 17 | sylbi 206 | . . . . . . . . . . . 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 242 | . . . . . . . . . 10 ⊢ (2 = 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
21 | 12, 20 | jaoi 393 | . . . . . . . . 9 ⊢ ((2 < 𝑁 ∨ 2 = 𝑁) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
22 | 10, 21 | syl6bi 242 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (2 ≤ 𝑁 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁)))) |
23 | 22 | imp 444 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 2 ≤ 𝑁) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
24 | 6, 23 | sylbi 206 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
25 | 4, 24 | jaoi 393 | . . . . 5 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
26 | 2, 25 | sylbi 206 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
27 | eleq1 2676 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
28 | 0noddALTV 40138 | . . . . . 6 ⊢ 0 ∉ Odd | |
29 | df-nel 2783 | . . . . . . 7 ⊢ (0 ∉ Odd ↔ ¬ 0 ∈ Odd ) | |
30 | pm2.21 119 | . . . . . . 7 ⊢ (¬ 0 ∈ Odd → (0 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) | |
31 | 29, 30 | sylbi 206 | . . . . . 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 242 | . . . 4 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
34 | 26, 33 | jaoi 393 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
35 | 1, 34 | sylbi 206 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ Odd → (𝑁 = 1 ∨ 2 < 𝑁))) |
36 | 35 | imp 444 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 class class class wbr 4583 ‘cfv 5804 ℝcr 9814 0cc0 9815 1c1 9816 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 Odd codd 40076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-even 40077 df-odd 40078 |
This theorem is referenced by: (None) |
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