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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 8750 |
. . 3
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2 | 1 | rabex 4080 |
. 2
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3 | elrabi 2841 |
. . . 4
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4 | 3 | peano2nnd 8759 |
. . 3
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5 | breq2 3941 |
. . . . . . 7
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6 | 5 | notbid 657 |
. . . . . 6
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7 | 6 | elrab 2844 |
. . . . 5
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8 | 7 | simprbi 273 |
. . . 4
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9 | 3 | nnzd 9196 |
. . . . 5
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10 | oddp1even 11609 |
. . . . 5
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11 | 9, 10 | syl 14 |
. . . 4
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12 | 8, 11 | mpbid 146 |
. . 3
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13 | nnehalf 11637 |
. . 3
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14 | 4, 12, 13 | syl2anc 409 |
. 2
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15 | nnz 9097 |
. . . . . 6
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16 | 2z 9106 |
. . . . . . 7
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17 | 16 | a1i 9 |
. . . . . 6
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18 | 15, 17 | zmulcld 9203 |
. . . . 5
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19 | peano2zm 9116 |
. . . . 5
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20 | 18, 19 | syl 14 |
. . . 4
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21 | 1e2m1 8863 |
. . . . 5
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22 | 17 | zred 9197 |
. . . . . 6
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23 | nnre 8751 |
. . . . . . 7
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24 | 23, 22 | remulcld 7820 |
. . . . . 6
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25 | 1red 7805 |
. . . . . 6
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26 | 0le2 8834 |
. . . . . . . 8
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27 | 26 | a1i 9 |
. . . . . . 7
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28 | nnge1 8767 |
. . . . . . 7
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29 | 22, 23, 27, 28 | lemulge12d 8720 |
. . . . . 6
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30 | 22, 24, 25, 29 | lesub1dd 8347 |
. . . . 5
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31 | 21, 30 | eqbrtrid 3971 |
. . . 4
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32 | elnnz1 9101 |
. . . 4
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33 | 20, 31, 32 | sylanbrc 414 |
. . 3
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34 | dvdsmul2 11552 |
. . . . 5
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35 | 15, 16, 34 | sylancl 410 |
. . . 4
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36 | oddm1even 11608 |
. . . . . 6
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37 | 18, 36 | syl 14 |
. . . . 5
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38 | 37 | biimprd 157 |
. . . 4
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39 | 35, 38 | mt2d 615 |
. . 3
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40 | breq2 3941 |
. . . . 5
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41 | 40 | notbid 657 |
. . . 4
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42 | 41 | elrab 2844 |
. . 3
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43 | 33, 39, 42 | sylanbrc 414 |
. 2
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44 | 3 | adantr 274 |
. . . . . . 7
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45 | 44 | nncnd 8758 |
. . . . . 6
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46 | 1cnd 7806 |
. . . . . 6
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47 | 45, 46 | addcld 7809 |
. . . . 5
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48 | simpr 109 |
. . . . . 6
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49 | 48 | nncnd 8758 |
. . . . 5
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50 | 2cnd 8817 |
. . . . 5
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51 | 2ap0 8837 |
. . . . . 6
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52 | 51 | a1i 9 |
. . . . 5
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53 | 47, 49, 50, 52 | divmulap3d 8609 |
. . . 4
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54 | 49, 50 | mulcld 7810 |
. . . . 5
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55 | 45, 46, 54 | addlsub 8156 |
. . . 4
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56 | 53, 55 | bitrd 187 |
. . 3
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57 | eqcom 2142 |
. . 3
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58 | 56, 57 | bitr3di 194 |
. 2
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59 | 2, 1, 14, 43, 58 | en3i 6673 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-xor 1355 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-en 6643 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079 df-dvds 11530 |
This theorem is referenced by: xpnnen 11943 unennn 11946 |
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