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Mirrors > Home > ILE Home > Th. List > nn0n0n1ge2 | GIF version |
Description: A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.) |
Ref | Expression |
---|---|
nn0n0n1ge2 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0cn 8744 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
2 | 1cnd 7565 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ) | |
3 | 1, 2, 2 | subsub4d 7885 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1))) |
4 | 1p1e2 8600 | . . . . . 6 ⊢ (1 + 1) = 2 | |
5 | 4 | oveq2i 5677 | . . . . 5 ⊢ (𝑁 − (1 + 1)) = (𝑁 − 2) |
6 | 3, 5 | syl6req 2138 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 2) = ((𝑁 − 1) − 1)) |
7 | 6 | 3ad2ant1 965 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (𝑁 − 2) = ((𝑁 − 1) − 1)) |
8 | 3simpa 941 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
9 | elnnne0 8748 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
10 | 8, 9 | sylibr 133 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 𝑁 ∈ ℕ) |
11 | nnm1nn0 8775 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
12 | 10, 11 | syl 14 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (𝑁 − 1) ∈ ℕ0) |
13 | 1, 2 | subeq0ad 7864 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) = 0 ↔ 𝑁 = 1)) |
14 | 13 | biimpd 143 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) = 0 → 𝑁 = 1)) |
15 | 14 | necon3d 2300 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ≠ 1 → (𝑁 − 1) ≠ 0)) |
16 | 15 | imp 123 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 1) → (𝑁 − 1) ≠ 0) |
17 | 16 | 3adant2 963 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (𝑁 − 1) ≠ 0) |
18 | elnnne0 8748 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ ↔ ((𝑁 − 1) ∈ ℕ0 ∧ (𝑁 − 1) ≠ 0)) | |
19 | 12, 17, 18 | sylanbrc 409 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (𝑁 − 1) ∈ ℕ) |
20 | nnm1nn0 8775 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → ((𝑁 − 1) − 1) ∈ ℕ0) | |
21 | 19, 20 | syl 14 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → ((𝑁 − 1) − 1) ∈ ℕ0) |
22 | 7, 21 | eqeltrd 2165 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (𝑁 − 2) ∈ ℕ0) |
23 | 2nn0 8751 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
24 | 23 | jctl 308 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) |
25 | 24 | 3ad2ant1 965 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) |
26 | nn0sub 8877 | . . 3 ⊢ ((2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2 ≤ 𝑁 ↔ (𝑁 − 2) ∈ ℕ0)) | |
27 | 25, 26 | syl 14 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → (2 ≤ 𝑁 ↔ (𝑁 − 2) ∈ ℕ0)) |
28 | 22, 27 | mpbird 166 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0 ∧ 𝑁 ≠ 1) → 2 ≤ 𝑁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 ≠ wne 2256 class class class wbr 3851 (class class class)co 5666 0cc0 7411 1c1 7412 + caddc 7414 ≤ cle 7584 − cmin 7714 ℕcn 8483 2c2 8534 ℕ0cn0 8734 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-inn 8484 df-2 8542 df-n0 8735 df-z 8812 |
This theorem is referenced by: nn0n0n1ge2b 8887 |
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