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Mirrors > Home > ILE Home > Th. List > nn1m1nn | GIF version |
Description: Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn1m1nn | ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 701 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ)) | |
2 | 1cnd 7782 | . . 3 ⊢ (𝑥 = 1 → 1 ∈ ℂ) | |
3 | 1, 2 | 2thd 174 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ 1 ∈ ℂ)) |
4 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | oveq1 5781 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 − 1) = (𝑦 − 1)) | |
6 | 5 | eleq1d 2208 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 − 1) ∈ ℕ ↔ (𝑦 − 1) ∈ ℕ)) |
7 | 4, 6 | orbi12d 782 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ))) |
8 | eqeq1 2146 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
9 | oveq1 5781 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 − 1) = ((𝑦 + 1) − 1)) | |
10 | 9 | eleq1d 2208 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 − 1) ∈ ℕ ↔ ((𝑦 + 1) − 1) ∈ ℕ)) |
11 | 8, 10 | orbi12d 782 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))) |
12 | eqeq1 2146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
13 | oveq1 5781 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1)) | |
14 | 13 | eleq1d 2208 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 − 1) ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ)) |
15 | 12, 14 | orbi12d 782 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))) |
16 | ax-1cn 7713 | . 2 ⊢ 1 ∈ ℂ | |
17 | nncn 8728 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | pncan 7968 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) − 1) = 𝑦) | |
19 | 17, 16, 18 | sylancl 409 | . . . . 5 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦) |
20 | id 19 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
21 | 19, 20 | eqeltrd 2216 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) ∈ ℕ) |
22 | 21 | olcd 723 | . . 3 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ)) |
23 | 22 | a1d 22 | . 2 ⊢ (𝑦 ∈ ℕ → ((𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ) → ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))) |
24 | 3, 7, 11, 15, 16, 23 | nnind 8736 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 697 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 1c1 7621 + caddc 7623 − cmin 7933 ℕcn 8720 |
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-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-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-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 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-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-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-inn 8721 |
This theorem is referenced by: nn1suc 8739 nnsub 8759 nnm1nn0 9018 nn0ge2m1nn 9037 |
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