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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 712 | . . 3 ⊢ (𝑥 = 1 → (𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ)) | |
2 | 1cnd 7970 | . . 3 ⊢ (𝑥 = 1 → 1 ∈ ℂ) | |
3 | 1, 2 | 2thd 175 | . 2 ⊢ (𝑥 = 1 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ 1 ∈ ℂ)) |
4 | eqeq1 2184 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1 ↔ 𝑦 = 1)) | |
5 | oveq1 5879 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 − 1) = (𝑦 − 1)) | |
6 | 5 | eleq1d 2246 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 − 1) ∈ ℕ ↔ (𝑦 − 1) ∈ ℕ)) |
7 | 4, 6 | orbi12d 793 | . 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝑦 = 1 ∨ (𝑦 − 1) ∈ ℕ))) |
8 | eqeq1 2184 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = 1 ↔ (𝑦 + 1) = 1)) | |
9 | oveq1 5879 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 − 1) = ((𝑦 + 1) − 1)) | |
10 | 9 | eleq1d 2246 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 − 1) ∈ ℕ ↔ ((𝑦 + 1) − 1) ∈ ℕ)) |
11 | 8, 10 | orbi12d 793 | . 2 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ ((𝑦 + 1) = 1 ∨ ((𝑦 + 1) − 1) ∈ ℕ))) |
12 | eqeq1 2184 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1 ↔ 𝐴 = 1)) | |
13 | oveq1 5879 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 − 1) = (𝐴 − 1)) | |
14 | 13 | eleq1d 2246 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 − 1) ∈ ℕ ↔ (𝐴 − 1) ∈ ℕ)) |
15 | 12, 14 | orbi12d 793 | . 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1 ∨ (𝑥 − 1) ∈ ℕ) ↔ (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ))) |
16 | ax-1cn 7901 | . 2 ⊢ 1 ∈ ℂ | |
17 | nncn 8923 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | pncan 8159 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑦 + 1) − 1) = 𝑦) | |
19 | 17, 16, 18 | sylancl 413 | . . . . 5 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) = 𝑦) |
20 | id 19 | . . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
21 | 19, 20 | eqeltrd 2254 | . . . 4 ⊢ (𝑦 ∈ ℕ → ((𝑦 + 1) − 1) ∈ ℕ) |
22 | 21 | olcd 734 | . . 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 8931 | 1 ⊢ (𝐴 ∈ ℕ → (𝐴 = 1 ∨ (𝐴 − 1) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 = wceq 1353 ∈ wcel 2148 (class class class)co 5872 ℂcc 7806 1c1 7809 + caddc 7811 − cmin 8124 ℕcn 8915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-sub 8126 df-inn 8916 |
This theorem is referenced by: nn1suc 8934 nnsub 8954 nnm1nn0 9213 nn0ge2m1nn 9232 |
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