Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elnn1uz2 | GIF version |
Description: A positive integer is either 1 or greater than or equal to 2. (Contributed by Paul Chapman, 17-Nov-2012.) |
Ref | Expression |
---|---|
elnn1uz2 | ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 700 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈ ℕ)) | |
2 | nnz 9073 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
3 | 1z 9080 | . . . . . . . 8 ⊢ 1 ∈ ℤ | |
4 | zdceq 9126 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ) → DECID 𝑁 = 1) | |
5 | 3, 4 | mpan2 421 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → DECID 𝑁 = 1) |
6 | df-dc 820 | . . . . . . 7 ⊢ (DECID 𝑁 = 1 ↔ (𝑁 = 1 ∨ ¬ 𝑁 = 1)) | |
7 | 5, 6 | sylib 121 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 = 1 ∨ ¬ 𝑁 = 1)) |
8 | df-ne 2309 | . . . . . . 7 ⊢ (𝑁 ≠ 1 ↔ ¬ 𝑁 = 1) | |
9 | 8 | orbi2i 751 | . . . . . 6 ⊢ ((𝑁 = 1 ∨ 𝑁 ≠ 1) ↔ (𝑁 = 1 ∨ ¬ 𝑁 = 1)) |
10 | 7, 9 | sylibr 133 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 = 1 ∨ 𝑁 ≠ 1)) |
11 | 2, 10 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ≠ 1)) |
12 | ordi 805 | . . . 4 ⊢ ((𝑁 = 1 ∨ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) ↔ ((𝑁 = 1 ∨ 𝑁 ∈ ℕ) ∧ (𝑁 = 1 ∨ 𝑁 ≠ 1))) | |
13 | 1, 11, 12 | sylanbrc 413 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))) |
14 | eluz2b3 9398 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1)) | |
15 | 14 | orbi2i 751 | . . 3 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) ↔ (𝑁 = 1 ∨ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))) |
16 | 13, 15 | sylibr 133 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) |
17 | 1nn 8731 | . . . 4 ⊢ 1 ∈ ℕ | |
18 | eleq1 2202 | . . . 4 ⊢ (𝑁 = 1 → (𝑁 ∈ ℕ ↔ 1 ∈ ℕ)) | |
19 | 17, 18 | mpbiri 167 | . . 3 ⊢ (𝑁 = 1 → 𝑁 ∈ ℕ) |
20 | eluz2nn 9364 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
21 | 19, 20 | jaoi 705 | . 2 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2)) → 𝑁 ∈ ℕ) |
22 | 16, 21 | impbii 125 | 1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∨ wo 697 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ‘cfv 5123 1c1 7621 ℕcn 8720 2c2 8771 ℤcz 9054 ℤ≥cuz 9326 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-uz 9327 |
This theorem is referenced by: indstr2 9403 dfphi2 11896 |
Copyright terms: Public domain | W3C validator |