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Mirrors > Home > ILE Home > Th. List > evenennn | GIF version |
Description: There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.) |
Ref | Expression |
---|---|
evenennn | ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 8694 | . . 3 ⊢ ℕ ∈ V | |
2 | 1 | rabex 4042 | . 2 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∈ V |
3 | breq2 3903 | . . . 4 ⊢ (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥)) | |
4 | 3 | elrab 2813 | . . 3 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ (𝑥 ∈ ℕ ∧ 2 ∥ 𝑥)) |
5 | nnehalf 11528 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 2 ∥ 𝑥) → (𝑥 / 2) ∈ ℕ) | |
6 | 4, 5 | sylbi 120 | . 2 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → (𝑥 / 2) ∈ ℕ) |
7 | 2nn 8849 | . . . . 5 ⊢ 2 ∈ ℕ | |
8 | 7 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℕ) |
9 | id 19 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
10 | 8, 9 | nnmulcld 8737 | . . 3 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℕ) |
11 | 2z 9050 | . . . 4 ⊢ 2 ∈ ℤ | |
12 | nnz 9041 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ) | |
13 | dvdsmul1 11442 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 2 ∥ (2 · 𝑦)) | |
14 | 11, 12, 13 | sylancr 410 | . . 3 ⊢ (𝑦 ∈ ℕ → 2 ∥ (2 · 𝑦)) |
15 | breq2 3903 | . . . 4 ⊢ (𝑧 = (2 · 𝑦) → (2 ∥ 𝑧 ↔ 2 ∥ (2 · 𝑦))) | |
16 | 15 | elrab 2813 | . . 3 ⊢ ((2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ ((2 · 𝑦) ∈ ℕ ∧ 2 ∥ (2 · 𝑦))) |
17 | 10, 14, 16 | sylanbrc 413 | . 2 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) |
18 | elrabi 2810 | . . . . . 6 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → 𝑥 ∈ ℕ) | |
19 | 18 | adantr 274 | . . . . 5 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℕ) |
20 | 19 | nncnd 8702 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℂ) |
21 | simpr 109 | . . . . 5 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ) | |
22 | 21 | nncnd 8702 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ) |
23 | 2cnd 8761 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ) | |
24 | 2ap0 8781 | . . . . 5 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 # 0) |
26 | 20, 22, 23, 25 | divmulap3d 8553 | . . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → ((𝑥 / 2) = 𝑦 ↔ 𝑥 = (𝑦 · 2))) |
27 | eqcom 2119 | . . . 4 ⊢ ((𝑥 / 2) = 𝑦 ↔ 𝑦 = (𝑥 / 2)) | |
28 | 27 | a1i 9 | . . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → ((𝑥 / 2) = 𝑦 ↔ 𝑦 = (𝑥 / 2))) |
29 | 22, 23 | mulcomd 7755 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑦 · 2) = (2 · 𝑦)) |
30 | 29 | eqeq2d 2129 | . . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑥 = (𝑦 · 2) ↔ 𝑥 = (2 · 𝑦))) |
31 | 26, 28, 30 | 3bitr3rd 218 | . 2 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑥 = (2 · 𝑦) ↔ 𝑦 = (𝑥 / 2))) |
32 | 2, 1, 6, 17, 31 | en3i 6633 | 1 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 {crab 2397 class class class wbr 3899 (class class class)co 5742 ≈ cen 6600 0cc0 7588 · cmul 7593 # cap 8311 / cdiv 8400 ℕcn 8688 2c2 8739 ℤcz 9022 ∥ cdvds 11420 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-en 6603 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 df-z 9023 df-dvds 11421 |
This theorem is referenced by: unennn 11837 |
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