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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnex 8956 |
. . 3
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2 | 1 | rabex 4162 |
. 2
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3 | elrabi 2905 |
. . . 4
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4 | 3 | peano2nnd 8965 |
. . 3
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5 | breq2 4022 |
. . . . . . 7
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6 | 5 | notbid 668 |
. . . . . 6
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7 | 6 | elrab 2908 |
. . . . 5
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8 | 7 | simprbi 275 |
. . . 4
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9 | 3 | nnzd 9405 |
. . . . 5
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10 | oddp1even 11916 |
. . . . 5
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11 | 9, 10 | syl 14 |
. . . 4
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12 | 8, 11 | mpbid 147 |
. . 3
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13 | nnehalf 11944 |
. . 3
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14 | 4, 12, 13 | syl2anc 411 |
. 2
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15 | nnz 9303 |
. . . . . 6
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16 | 2z 9312 |
. . . . . . 7
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17 | 16 | a1i 9 |
. . . . . 6
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18 | 15, 17 | zmulcld 9412 |
. . . . 5
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19 | peano2zm 9322 |
. . . . 5
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20 | 18, 19 | syl 14 |
. . . 4
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21 | 1e2m1 9069 |
. . . . 5
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22 | 17 | zred 9406 |
. . . . . 6
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23 | nnre 8957 |
. . . . . . 7
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24 | 23, 22 | remulcld 8019 |
. . . . . 6
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25 | 1red 8003 |
. . . . . 6
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26 | 0le2 9040 |
. . . . . . . 8
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27 | 26 | a1i 9 |
. . . . . . 7
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28 | nnge1 8973 |
. . . . . . 7
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29 | 22, 23, 27, 28 | lemulge12d 8926 |
. . . . . 6
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30 | 22, 24, 25, 29 | lesub1dd 8549 |
. . . . 5
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31 | 21, 30 | eqbrtrid 4053 |
. . . 4
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32 | elnnz1 9307 |
. . . 4
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33 | 20, 31, 32 | sylanbrc 417 |
. . 3
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34 | dvdsmul2 11856 |
. . . . 5
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35 | 15, 16, 34 | sylancl 413 |
. . . 4
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36 | oddm1even 11915 |
. . . . . 6
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37 | 18, 36 | syl 14 |
. . . . 5
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38 | 37 | biimprd 158 |
. . . 4
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39 | 35, 38 | mt2d 626 |
. . 3
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40 | breq2 4022 |
. . . . 5
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41 | 40 | notbid 668 |
. . . 4
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42 | 41 | elrab 2908 |
. . 3
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43 | 33, 39, 42 | sylanbrc 417 |
. 2
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44 | 3 | adantr 276 |
. . . . . . 7
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45 | 44 | nncnd 8964 |
. . . . . 6
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46 | 1cnd 8004 |
. . . . . 6
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47 | 45, 46 | addcld 8008 |
. . . . 5
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48 | simpr 110 |
. . . . . 6
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49 | 48 | nncnd 8964 |
. . . . 5
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50 | 2cnd 9023 |
. . . . 5
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51 | 2ap0 9043 |
. . . . . 6
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52 | 51 | a1i 9 |
. . . . 5
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53 | 47, 49, 50, 52 | divmulap3d 8813 |
. . . 4
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54 | 49, 50 | mulcld 8009 |
. . . . 5
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55 | 45, 46, 54 | addlsub 8358 |
. . . 4
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56 | 53, 55 | bitrd 188 |
. . 3
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57 | eqcom 2191 |
. . 3
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58 | 56, 57 | bitr3di 195 |
. 2
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59 | 2, 1, 14, 43, 58 | en3i 6798 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-en 6768 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-n0 9208 df-z 9285 df-dvds 11830 |
This theorem is referenced by: xpnnen 12448 unennn 12451 |
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