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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 684 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ)) | |
2 | 1cnd 7706 | . . 3 ⊢ (𝑥 = 1 → 1 ∈ ℂ) | |
3 | 1, 2 | 2thd 174 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ 1 ∈ ℂ)) |
4 | eqeq1 2121 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | oveq1 5735 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 − 1) = (𝑦 − 1)) | |
6 | 5 | eleq1d 2183 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 − 1) ∈ ℕ ↔ (𝑦 − 1) ∈ ℕ)) |
7 | 4, 6 | orbi12d 765 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ))) |
8 | eqeq1 2121 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
9 | oveq1 5735 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 − 1) = ((𝑦 + 1) − 1)) | |
10 | 9 | eleq1d 2183 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 − 1) ∈ ℕ ↔ ((𝑦 + 1) − 1) ∈ ℕ)) |
11 | 8, 10 | orbi12d 765 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))) |
12 | eqeq1 2121 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
13 | oveq1 5735 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1)) | |
14 | 13 | eleq1d 2183 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 − 1) ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ)) |
15 | 12, 14 | orbi12d 765 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))) |
16 | ax-1cn 7638 | . 2 ⊢ 1 ∈ ℂ | |
17 | nncn 8638 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | pncan 7891 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) − 1) = 𝑦) | |
19 | 17, 16, 18 | sylancl 407 | . . . . 5 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦) |
20 | id 19 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
21 | 19, 20 | eqeltrd 2191 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) ∈ ℕ) |
22 | 21 | olcd 706 | . . 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 8646 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 680 = wceq 1314 ∈ wcel 1463 (class class class)co 5728 ℂcc 7545 1c1 7548 + caddc 7550 − cmin 7856 ℕcn 8630 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-addass 7647 ax-distr 7649 ax-i2m1 7650 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-sub 7858 df-inn 8631 |
This theorem is referenced by: nn1suc 8649 nnsub 8669 nnm1nn0 8922 nn0ge2m1nn 8941 |
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