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Mirrors > Home > ILE Home > Th. List > nn0le2is012 | GIF version |
Description: A nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 16-Mar-2019.) |
Ref | Expression |
---|---|
nn0le2is012 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 9098 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | 2z 9106 | . . . 4 ⊢ 2 ∈ ℤ | |
3 | zleloe 9125 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑁 ≤ 2 ↔ (𝑁 < 2 ∨ 𝑁 = 2))) | |
4 | 1, 2, 3 | sylancl 410 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 2 ↔ (𝑁 < 2 ∨ 𝑁 = 2))) |
5 | zltlem1 9135 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑁 < 2 ↔ 𝑁 ≤ (2 − 1))) | |
6 | 1, 2, 5 | sylancl 410 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 2 ↔ 𝑁 ≤ (2 − 1))) |
7 | 2m1e1 8862 | . . . . . . . . . 10 ⊢ (2 − 1) = 1 | |
8 | 7 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (2 − 1) = 1) |
9 | 8 | breq2d 3949 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ (2 − 1) ↔ 𝑁 ≤ 1)) |
10 | 6, 9 | bitrd 187 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 2 ↔ 𝑁 ≤ 1)) |
11 | 1z 9104 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
12 | zleloe 9125 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → (𝑁 ≤ 1 ↔ (𝑁 < 1 ∨ 𝑁 = 1))) | |
13 | 1, 11, 12 | sylancl 410 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 1 ↔ (𝑁 < 1 ∨ 𝑁 = 1))) |
14 | nn0lt10b 9155 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 ↔ 𝑁 = 0)) | |
15 | 3mix1 1151 | . . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) | |
16 | 14, 15 | syl6bi 162 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 1 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
17 | 16 | com12 30 | . . . . . . . . . 10 ⊢ (𝑁 < 1 → (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
18 | 3mix2 1152 | . . . . . . . . . . 11 ⊢ (𝑁 = 1 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) | |
19 | 18 | a1d 22 | . . . . . . . . . 10 ⊢ (𝑁 = 1 → (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
20 | 17, 19 | jaoi 706 | . . . . . . . . 9 ⊢ ((𝑁 < 1 ∨ 𝑁 = 1) → (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
21 | 20 | com12 30 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 < 1 ∨ 𝑁 = 1) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
22 | 13, 21 | sylbid 149 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 1 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
23 | 10, 22 | sylbid 149 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 < 2 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
24 | 23 | com12 30 | . . . . 5 ⊢ (𝑁 < 2 → (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
25 | 3mix3 1153 | . . . . . 6 ⊢ (𝑁 = 2 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) | |
26 | 25 | a1d 22 | . . . . 5 ⊢ (𝑁 = 2 → (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
27 | 24, 26 | jaoi 706 | . . . 4 ⊢ ((𝑁 < 2 ∨ 𝑁 = 2) → (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
28 | 27 | com12 30 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 < 2 ∨ 𝑁 = 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
29 | 4, 28 | sylbid 149 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≤ 2 → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2))) |
30 | 29 | imp 123 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∨ w3o 962 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 0cc0 7644 1c1 7645 < clt 7824 ≤ cle 7825 − cmin 7957 2c2 8795 ℕ0cn0 9001 ℤcz 9078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079 |
This theorem is referenced by: xnn0le2is012 9679 |
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