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Mirrors > Home > ILE Home > Th. List > 1idssfct | GIF version |
Description: The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
1idssfct | ⊢ (𝑁 ∈ ℕ → {1, 𝑁} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 8876 | . . 3 ⊢ 1 ∈ ℕ | |
2 | nnz 9218 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
3 | 1dvds 11754 | . . . 4 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℕ → 1 ∥ 𝑁) |
5 | breq1 3990 | . . . . 5 ⊢ (𝑛 = 1 → (𝑛 ∥ 𝑁 ↔ 1 ∥ 𝑁)) | |
6 | 5 | elrab 2886 | . . . 4 ⊢ (1 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ↔ (1 ∈ ℕ ∧ 1 ∥ 𝑁)) |
7 | 6 | biimpri 132 | . . 3 ⊢ ((1 ∈ ℕ ∧ 1 ∥ 𝑁) → 1 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) |
8 | 1, 4, 7 | sylancr 412 | . 2 ⊢ (𝑁 ∈ ℕ → 1 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) |
9 | iddvds 11753 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∥ 𝑁) | |
10 | 2, 9 | syl 14 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∥ 𝑁) |
11 | breq1 3990 | . . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛 ∥ 𝑁 ↔ 𝑁 ∥ 𝑁)) | |
12 | 11 | elrab 2886 | . . . 4 ⊢ (𝑁 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ↔ (𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑁)) |
13 | 12 | biimpri 132 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ∥ 𝑁) → 𝑁 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) |
14 | 10, 13 | mpdan 419 | . 2 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) |
15 | prssi 3736 | . 2 ⊢ ((1 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁} ∧ 𝑁 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) → {1, 𝑁} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) | |
16 | 8, 14, 15 | syl2anc 409 | 1 ⊢ (𝑁 ∈ ℕ → {1, 𝑁} ⊆ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑁}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 {crab 2452 ⊆ wss 3121 {cpr 3582 class class class wbr 3987 1c1 7762 ℕcn 8865 ℤcz 9199 ∥ cdvds 11736 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-z 9200 df-dvds 11737 |
This theorem is referenced by: isprm2 12058 |
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