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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 8923 | . . 3 ⊢ ℕ ∈ V | |
2 | 1 | rabex 4147 | . 2 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∈ V |
3 | breq2 4007 | . . . 4 ⊢ (𝑧 = 𝑥 → (2 ∥ 𝑧 ↔ 2 ∥ 𝑥)) | |
4 | 3 | elrab 2893 | . . 3 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ (𝑥 ∈ ℕ ∧ 2 ∥ 𝑥)) |
5 | nnehalf 11903 | . . 3 ⊢ ((𝑥 ∈ ℕ ∧ 2 ∥ 𝑥) → (𝑥 / 2) ∈ ℕ) | |
6 | 4, 5 | sylbi 121 | . 2 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → (𝑥 / 2) ∈ ℕ) |
7 | 2nn 9078 | . . . . 5 ⊢ 2 ∈ ℕ | |
8 | 7 | a1i 9 | . . . 4 ⊢ (𝑦 ∈ ℕ → 2 ∈ ℕ) |
9 | id 19 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ) | |
10 | 8, 9 | nnmulcld 8966 | . . 3 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ ℕ) |
11 | 2z 9279 | . . . 4 ⊢ 2 ∈ ℤ | |
12 | nnz 9270 | . . . 4 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ) | |
13 | dvdsmul1 11815 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 2 ∥ (2 · 𝑦)) | |
14 | 11, 12, 13 | sylancr 414 | . . 3 ⊢ (𝑦 ∈ ℕ → 2 ∥ (2 · 𝑦)) |
15 | breq2 4007 | . . . 4 ⊢ (𝑧 = (2 · 𝑦) → (2 ∥ 𝑧 ↔ 2 ∥ (2 · 𝑦))) | |
16 | 15 | elrab 2893 | . . 3 ⊢ ((2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ↔ ((2 · 𝑦) ∈ ℕ ∧ 2 ∥ (2 · 𝑦))) |
17 | 10, 14, 16 | sylanbrc 417 | . 2 ⊢ (𝑦 ∈ ℕ → (2 · 𝑦) ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧}) |
18 | elrabi 2890 | . . . . . 6 ⊢ (𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} → 𝑥 ∈ ℕ) | |
19 | 18 | adantr 276 | . . . . 5 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℕ) |
20 | 19 | nncnd 8931 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℂ) |
21 | simpr 110 | . . . . 5 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ) | |
22 | 21 | nncnd 8931 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ) |
23 | 2cnd 8990 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 ∈ ℂ) | |
24 | 2ap0 9010 | . . . . 5 ⊢ 2 # 0 | |
25 | 24 | a1i 9 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → 2 # 0) |
26 | 20, 22, 23, 25 | divmulap3d 8780 | . . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → ((𝑥 / 2) = 𝑦 ↔ 𝑥 = (𝑦 · 2))) |
27 | eqcom 2179 | . . . 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 7977 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑦 · 2) = (2 · 𝑦)) |
30 | 29 | eqeq2d 2189 | . . 3 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑥 = (𝑦 · 2) ↔ 𝑥 = (2 · 𝑦))) |
31 | 26, 28, 30 | 3bitr3rd 219 | . 2 ⊢ ((𝑥 ∈ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ∧ 𝑦 ∈ ℕ) → (𝑥 = (2 · 𝑦) ↔ 𝑦 = (𝑥 / 2))) |
32 | 2, 1, 6, 17, 31 | en3i 6770 | 1 ⊢ {𝑧 ∈ ℕ ∣ 2 ∥ 𝑧} ≈ ℕ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {crab 2459 class class class wbr 4003 (class class class)co 5874 ≈ cen 6737 0cc0 7810 · cmul 7815 # cap 8536 / cdiv 8627 ℕcn 8917 2c2 8968 ℤcz 9251 ∥ cdvds 11789 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3or 979 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-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 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 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-en 6740 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-n0 9175 df-z 9252 df-dvds 11790 |
This theorem is referenced by: unennn 12392 |
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