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Mirrors > Home > ILE Home > Th. List > oddennn | Unicode version |
Description: There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.) |
Ref | Expression |
---|---|
oddennn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 8690 | . . 3 | |
2 | 1 | rabex 4042 | . 2 |
3 | elrabi 2810 | . . . 4 | |
4 | 3 | peano2nnd 8699 | . . 3 |
5 | breq2 3903 | . . . . . . 7 | |
6 | 5 | notbid 641 | . . . . . 6 |
7 | 6 | elrab 2813 | . . . . 5 |
8 | 7 | simprbi 273 | . . . 4 |
9 | 3 | nnzd 9130 | . . . . 5 |
10 | oddp1even 11485 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 8, 11 | mpbid 146 | . . 3 |
13 | nnehalf 11513 | . . 3 | |
14 | 4, 12, 13 | syl2anc 408 | . 2 |
15 | nnz 9031 | . . . . . 6 | |
16 | 2z 9040 | . . . . . . 7 | |
17 | 16 | a1i 9 | . . . . . 6 |
18 | 15, 17 | zmulcld 9137 | . . . . 5 |
19 | peano2zm 9050 | . . . . 5 | |
20 | 18, 19 | syl 14 | . . . 4 |
21 | 1e2m1 8803 | . . . . 5 | |
22 | 17 | zred 9131 | . . . . . 6 |
23 | nnre 8691 | . . . . . . 7 | |
24 | 23, 22 | remulcld 7764 | . . . . . 6 |
25 | 1red 7749 | . . . . . 6 | |
26 | 0le2 8774 | . . . . . . . 8 | |
27 | 26 | a1i 9 | . . . . . . 7 |
28 | nnge1 8707 | . . . . . . 7 | |
29 | 22, 23, 27, 28 | lemulge12d 8660 | . . . . . 6 |
30 | 22, 24, 25, 29 | lesub1dd 8290 | . . . . 5 |
31 | 21, 30 | eqbrtrid 3933 | . . . 4 |
32 | elnnz1 9035 | . . . 4 | |
33 | 20, 31, 32 | sylanbrc 413 | . . 3 |
34 | dvdsmul2 11428 | . . . . 5 | |
35 | 15, 16, 34 | sylancl 409 | . . . 4 |
36 | oddm1even 11484 | . . . . . 6 | |
37 | 18, 36 | syl 14 | . . . . 5 |
38 | 37 | biimprd 157 | . . . 4 |
39 | 35, 38 | mt2d 599 | . . 3 |
40 | breq2 3903 | . . . . 5 | |
41 | 40 | notbid 641 | . . . 4 |
42 | 41 | elrab 2813 | . . 3 |
43 | 33, 39, 42 | sylanbrc 413 | . 2 |
44 | eqcom 2119 | . . 3 | |
45 | 3 | adantr 274 | . . . . . . 7 |
46 | 45 | nncnd 8698 | . . . . . 6 |
47 | 1cnd 7750 | . . . . . 6 | |
48 | 46, 47 | addcld 7753 | . . . . 5 |
49 | simpr 109 | . . . . . 6 | |
50 | 49 | nncnd 8698 | . . . . 5 |
51 | 2cnd 8757 | . . . . 5 | |
52 | 2ap0 8777 | . . . . . 6 # | |
53 | 52 | a1i 9 | . . . . 5 # |
54 | 48, 50, 51, 53 | divmulap3d 8552 | . . . 4 |
55 | 50, 51 | mulcld 7754 | . . . . 5 |
56 | 46, 47, 55 | addlsub 8100 | . . . 4 |
57 | 54, 56 | bitrd 187 | . . 3 |
58 | 44, 57 | syl5rbbr 194 | . 2 |
59 | 2, 1, 14, 43, 58 | en3i 6633 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wceq 1316 wcel 1465 crab 2397 class class class wbr 3899 (class class class)co 5742 cen 6600 cc0 7588 c1 7589 caddc 7591 cmul 7593 cle 7769 cmin 7901 # cap 8310 cdiv 8399 cn 8684 c2 8735 cz 9012 cdvds 11405 |
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-xor 1339 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 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-n0 8936 df-z 9013 df-dvds 11406 |
This theorem is referenced by: xpnnen 11818 unennn 11821 |
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