Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8854 | . . 3 ⊢ ℕ ∈ V | |
2 | 1 | rabex 4120 | . 2 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∈ V |
3 | breq2 3980 | . . . 4 ⊢ (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥)) | |
4 | 3 | elrab 2877 | . . 3 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ (𝑥 ∈ ℕ ∧ 2 ∥ 𝑥)) |
5 | nnehalf 11826 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 2 ∥ 𝑥) → (𝑥 / 2) ∈ ℕ) | |
6 | 4, 5 | sylbi 120 | . 2 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → (𝑥 / 2) ∈ ℕ) |
7 | 2nn 9009 | . . . . 5 ⊢ 2 ∈ ℕ | |
8 | 7 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℕ) |
9 | id 19 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
10 | 8, 9 | nnmulcld 8897 | . . 3 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℕ) |
11 | 2z 9210 | . . . 4 ⊢ 2 ∈ ℤ | |
12 | nnz 9201 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ) | |
13 | dvdsmul1 11739 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 2 ∥ (2 · 𝑦)) | |
14 | 11, 12, 13 | sylancr 411 | . . 3 ⊢ (𝑦 ∈ ℕ → 2 ∥ (2 · 𝑦)) |
15 | breq2 3980 | . . . 4 ⊢ (𝑧 = (2 · 𝑦) → (2 ∥ 𝑧 ↔ 2 ∥ (2 · 𝑦))) | |
16 | 15 | elrab 2877 | . . 3 ⊢ ((2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ ((2 · 𝑦) ∈ ℕ ∧ 2 ∥ (2 · 𝑦))) |
17 | 10, 14, 16 | sylanbrc 414 | . 2 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) |
18 | elrabi 2874 | . . . . . 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 8862 | . . . 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 8862 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ) |
23 | 2cnd 8921 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ) | |
24 | 2ap0 8941 | . . . . 5 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 # 0) |
26 | 20, 22, 23, 25 | divmulap3d 8712 | . . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → ((𝑥 / 2) = 𝑦 ↔ 𝑥 = (𝑦 · 2))) |
27 | eqcom 2166 | . . . 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 7911 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑦 · 2) = (2 · 𝑦)) |
30 | 29 | eqeq2d 2176 | . . 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 6728 | 1 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 {crab 2446 class class class wbr 3976 (class class class)co 5836 ≈ cen 6695 0cc0 7744 · cmul 7749 # cap 8470 / cdiv 8559 ℕcn 8848 2c2 8899 ℤcz 9182 ∥ cdvds 11713 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-en 6698 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-n0 9106 df-z 9183 df-dvds 11714 |
This theorem is referenced by: unennn 12267 |
Copyright terms: Public domain | W3C validator |