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Mirrors > Home > ILE Home > Th. List > nn0lt2 | GIF version |
Description: A nonnegative integer less than 2 must be 0 or 1. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
Ref | Expression |
---|---|
nn0lt2 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 = 0 ∨ 𝑁 = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 711 | . . 3 ⊢ (𝑁 = 1 → (𝑁 = 0 ∨ 𝑁 = 1)) | |
2 | 1 | a1i 9 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 = 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
3 | nn0z 9249 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
4 | 2z 9257 | . . . . . 6 ⊢ 2 ∈ ℤ | |
5 | zltlem1 9286 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑁 < 2 ↔ 𝑁 ≤ (2 − 1))) | |
6 | 3, 4, 5 | sylancl 413 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 2 ↔ 𝑁 ≤ (2 − 1))) |
7 | 2m1e1 9013 | . . . . . 6 ⊢ (2 − 1) = 1 | |
8 | 7 | breq2i 4008 | . . . . 5 ⊢ (𝑁 ≤ (2 − 1) ↔ 𝑁 ≤ 1) |
9 | 6, 8 | bitrdi 196 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 2 ↔ 𝑁 ≤ 1)) |
10 | necom 2431 | . . . . 5 ⊢ (𝑁 ≠ 1 ↔ 1 ≠ 𝑁) | |
11 | 1z 9255 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
12 | zltlen 9307 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 < 1 ↔ (𝑁 ≤ 1 ∧ 1 ≠ 𝑁))) | |
13 | 3, 11, 12 | sylancl 413 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ (𝑁 ≤ 1 ∧ 1 ≠ 𝑁))) |
14 | nn0lt10b 9309 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0)) | |
15 | 14 | biimpa 296 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 1) → 𝑁 = 0) |
16 | 15 | orcd 733 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 1) → (𝑁 = 0 ∨ 𝑁 = 1)) |
17 | 16 | ex 115 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
18 | 13, 17 | sylbird 170 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 ≤ 1 ∧ 1 ≠ 𝑁) → (𝑁 = 0 ∨ 𝑁 = 1))) |
19 | 18 | expd 258 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 1 → (1 ≠ 𝑁 → (𝑁 = 0 ∨ 𝑁 = 1)))) |
20 | 10, 19 | syl7bi 165 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 1 → (𝑁 ≠ 1 → (𝑁 = 0 ∨ 𝑁 = 1)))) |
21 | 9, 20 | sylbid 150 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 2 → (𝑁 ≠ 1 → (𝑁 = 0 ∨ 𝑁 = 1)))) |
22 | 21 | imp 124 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 ≠ 1 → (𝑁 = 0 ∨ 𝑁 = 1))) |
23 | zdceq 9304 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → DECID 𝑁 = 1) | |
24 | 3, 11, 23 | sylancl 413 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → DECID 𝑁 = 1) |
25 | 24 | adantr 276 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → DECID 𝑁 = 1) |
26 | dcne 2358 | . . 3 ⊢ (DECID 𝑁 = 1 ↔ (𝑁 = 1 ∨ 𝑁 ≠ 1)) | |
27 | 25, 26 | sylib 122 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 = 1 ∨ 𝑁 ≠ 1)) |
28 | 2, 22, 27 | mpjaod 718 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 < 2) → (𝑁 = 0 ∨ 𝑁 = 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4000 (class class class)co 5868 0cc0 7789 1c1 7790 < clt 7969 ≤ cle 7970 − cmin 8105 2c2 8946 ℕ0cn0 9152 ℤcz 9229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-inn 8896 df-2 8954 df-n0 9153 df-z 9230 |
This theorem is referenced by: (None) |
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