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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 8877 | . . 3 | |
2 | 1 | rabex 4131 | . 2 |
3 | elrabi 2883 | . . . 4 | |
4 | 3 | peano2nnd 8886 | . . 3 |
5 | breq2 3991 | . . . . . . 7 | |
6 | 5 | notbid 662 | . . . . . 6 |
7 | 6 | elrab 2886 | . . . . 5 |
8 | 7 | simprbi 273 | . . . 4 |
9 | 3 | nnzd 9326 | . . . . 5 |
10 | oddp1even 11828 | . . . . 5 | |
11 | 9, 10 | syl 14 | . . . 4 |
12 | 8, 11 | mpbid 146 | . . 3 |
13 | nnehalf 11856 | . . 3 | |
14 | 4, 12, 13 | syl2anc 409 | . 2 |
15 | nnz 9224 | . . . . . 6 | |
16 | 2z 9233 | . . . . . . 7 | |
17 | 16 | a1i 9 | . . . . . 6 |
18 | 15, 17 | zmulcld 9333 | . . . . 5 |
19 | peano2zm 9243 | . . . . 5 | |
20 | 18, 19 | syl 14 | . . . 4 |
21 | 1e2m1 8990 | . . . . 5 | |
22 | 17 | zred 9327 | . . . . . 6 |
23 | nnre 8878 | . . . . . . 7 | |
24 | 23, 22 | remulcld 7943 | . . . . . 6 |
25 | 1red 7928 | . . . . . 6 | |
26 | 0le2 8961 | . . . . . . . 8 | |
27 | 26 | a1i 9 | . . . . . . 7 |
28 | nnge1 8894 | . . . . . . 7 | |
29 | 22, 23, 27, 28 | lemulge12d 8847 | . . . . . 6 |
30 | 22, 24, 25, 29 | lesub1dd 8473 | . . . . 5 |
31 | 21, 30 | eqbrtrid 4022 | . . . 4 |
32 | elnnz1 9228 | . . . 4 | |
33 | 20, 31, 32 | sylanbrc 415 | . . 3 |
34 | dvdsmul2 11769 | . . . . 5 | |
35 | 15, 16, 34 | sylancl 411 | . . . 4 |
36 | oddm1even 11827 | . . . . . 6 | |
37 | 18, 36 | syl 14 | . . . . 5 |
38 | 37 | biimprd 157 | . . . 4 |
39 | 35, 38 | mt2d 620 | . . 3 |
40 | breq2 3991 | . . . . 5 | |
41 | 40 | notbid 662 | . . . 4 |
42 | 41 | elrab 2886 | . . 3 |
43 | 33, 39, 42 | sylanbrc 415 | . 2 |
44 | 3 | adantr 274 | . . . . . . 7 |
45 | 44 | nncnd 8885 | . . . . . 6 |
46 | 1cnd 7929 | . . . . . 6 | |
47 | 45, 46 | addcld 7932 | . . . . 5 |
48 | simpr 109 | . . . . . 6 | |
49 | 48 | nncnd 8885 | . . . . 5 |
50 | 2cnd 8944 | . . . . 5 | |
51 | 2ap0 8964 | . . . . . 6 # | |
52 | 51 | a1i 9 | . . . . 5 # |
53 | 47, 49, 50, 52 | divmulap3d 8735 | . . . 4 |
54 | 49, 50 | mulcld 7933 | . . . . 5 |
55 | 45, 46, 54 | addlsub 8282 | . . . 4 |
56 | 53, 55 | bitrd 187 | . . 3 |
57 | eqcom 2172 | . . 3 | |
58 | 56, 57 | bitr3di 194 | . 2 |
59 | 2, 1, 14, 43, 58 | en3i 6747 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wceq 1348 wcel 2141 crab 2452 class class class wbr 3987 (class class class)co 5851 cen 6714 cc0 7767 c1 7768 caddc 7770 cmul 7772 cle 7948 cmin 8083 # cap 8493 cdiv 8582 cn 8871 c2 8922 cz 9205 cdvds 11742 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-xor 1371 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-en 6717 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-n0 9129 df-z 9206 df-dvds 11743 |
This theorem is referenced by: xpnnen 12342 unennn 12345 |
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