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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 8726 | . . 3 ⊢ ℕ ∈ V | |
2 | 1 | rabex 4072 | . 2 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∈ V |
3 | breq2 3933 | . . . 4 ⊢ (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥)) | |
4 | 3 | elrab 2840 | . . 3 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ (𝑥 ∈ ℕ ∧ 2 ∥ 𝑥)) |
5 | nnehalf 11601 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 2 ∥ 𝑥) → (𝑥 / 2) ∈ ℕ) | |
6 | 4, 5 | sylbi 120 | . 2 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → (𝑥 / 2) ∈ ℕ) |
7 | 2nn 8881 | . . . . 5 ⊢ 2 ∈ ℕ | |
8 | 7 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℕ) |
9 | id 19 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
10 | 8, 9 | nnmulcld 8769 | . . 3 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℕ) |
11 | 2z 9082 | . . . 4 ⊢ 2 ∈ ℤ | |
12 | nnz 9073 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ) | |
13 | dvdsmul1 11515 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 2 ∥ (2 · 𝑦)) | |
14 | 11, 12, 13 | sylancr 410 | . . 3 ⊢ (𝑦 ∈ ℕ → 2 ∥ (2 · 𝑦)) |
15 | breq2 3933 | . . . 4 ⊢ (𝑧 = (2 · 𝑦) → (2 ∥ 𝑧 ↔ 2 ∥ (2 · 𝑦))) | |
16 | 15 | elrab 2840 | . . 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 2837 | . . . . . 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 8734 | . . . 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 8734 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ) |
23 | 2cnd 8793 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ) | |
24 | 2ap0 8813 | . . . . 5 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 # 0) |
26 | 20, 22, 23, 25 | divmulap3d 8585 | . . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → ((𝑥 / 2) = 𝑦 ↔ 𝑥 = (𝑦 · 2))) |
27 | eqcom 2141 | . . . 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 7787 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑦 · 2) = (2 · 𝑦)) |
30 | 29 | eqeq2d 2151 | . . 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 6665 | 1 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {crab 2420 class class class wbr 3929 (class class class)co 5774 ≈ cen 6632 0cc0 7620 · cmul 7625 # cap 8343 / cdiv 8432 ℕcn 8720 2c2 8771 ℤcz 9054 ∥ cdvds 11493 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-en 6635 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 df-dvds 11494 |
This theorem is referenced by: unennn 11910 |
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