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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 702 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ)) | |
2 | 1cnd 7906 | . . 3 ⊢ (𝑥 = 1 → 1 ∈ ℂ) | |
3 | 1, 2 | 2thd 174 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ 1 ∈ ℂ)) |
4 | eqeq1 2171 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | oveq1 5843 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 − 1) = (𝑦 − 1)) | |
6 | 5 | eleq1d 2233 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 − 1) ∈ ℕ ↔ (𝑦 − 1) ∈ ℕ)) |
7 | 4, 6 | orbi12d 783 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ))) |
8 | eqeq1 2171 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
9 | oveq1 5843 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 − 1) = ((𝑦 + 1) − 1)) | |
10 | 9 | eleq1d 2233 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 − 1) ∈ ℕ ↔ ((𝑦 + 1) − 1) ∈ ℕ)) |
11 | 8, 10 | orbi12d 783 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))) |
12 | eqeq1 2171 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
13 | oveq1 5843 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1)) | |
14 | 13 | eleq1d 2233 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 − 1) ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ)) |
15 | 12, 14 | orbi12d 783 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))) |
16 | ax-1cn 7837 | . 2 ⊢ 1 ∈ ℂ | |
17 | nncn 8856 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | pncan 8095 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) − 1) = 𝑦) | |
19 | 17, 16, 18 | sylancl 410 | . . . . 5 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦) |
20 | id 19 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
21 | 19, 20 | eqeltrd 2241 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) ∈ ℕ) |
22 | 21 | olcd 724 | . . 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 8864 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 = wceq 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 1c1 7745 + caddc 7747 − cmin 8060 ℕcn 8848 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-inn 8849 |
This theorem is referenced by: nn1suc 8867 nnsub 8887 nnm1nn0 9146 nn0ge2m1nn 9165 |
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