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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 8584 |
. . 3
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2 | 1 | rabex 4012 |
. 2
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3 | elrabi 2790 |
. . . 4
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4 | 3 | peano2nnd 8593 |
. . 3
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5 | breq2 3879 |
. . . . . . 7
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6 | 5 | notbid 633 |
. . . . . 6
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7 | 6 | elrab 2793 |
. . . . 5
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8 | 7 | simprbi 271 |
. . . 4
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9 | 3 | nnzd 9024 |
. . . . 5
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10 | oddp1even 11368 |
. . . . 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 11396 |
. . 3
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14 | 4, 12, 13 | syl2anc 406 |
. 2
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15 | nnz 8925 |
. . . . . 6
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16 | 2z 8934 |
. . . . . . 7
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17 | 16 | a1i 9 |
. . . . . 6
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18 | 15, 17 | zmulcld 9031 |
. . . . 5
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19 | peano2zm 8944 |
. . . . 5
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20 | 18, 19 | syl 14 |
. . . 4
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21 | 1e2m1 8697 |
. . . . 5
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22 | 17 | zred 9025 |
. . . . . 6
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23 | nnre 8585 |
. . . . . . 7
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24 | 23, 22 | remulcld 7668 |
. . . . . 6
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25 | 1red 7653 |
. . . . . 6
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26 | 0le2 8668 |
. . . . . . . 8
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27 | 26 | a1i 9 |
. . . . . . 7
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28 | nnge1 8601 |
. . . . . . 7
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29 | 22, 23, 27, 28 | lemulge12d 8554 |
. . . . . 6
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30 | 22, 24, 25, 29 | lesub1dd 8189 |
. . . . 5
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31 | 21, 30 | syl5eqbr 3908 |
. . . 4
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32 | elnnz1 8929 |
. . . 4
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33 | 20, 31, 32 | sylanbrc 411 |
. . 3
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34 | dvdsmul2 11311 |
. . . . 5
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35 | 15, 16, 34 | sylancl 407 |
. . . 4
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36 | oddm1even 11367 |
. . . . . 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 595 |
. . 3
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40 | breq2 3879 |
. . . . 5
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41 | 40 | notbid 633 |
. . . 4
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42 | 41 | elrab 2793 |
. . 3
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43 | 33, 39, 42 | sylanbrc 411 |
. 2
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44 | eqcom 2102 |
. . 3
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45 | 3 | adantr 272 |
. . . . . . 7
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46 | 45 | nncnd 8592 |
. . . . . 6
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47 | 1cnd 7654 |
. . . . . 6
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48 | 46, 47 | addcld 7657 |
. . . . 5
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49 | simpr 109 |
. . . . . 6
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50 | 49 | nncnd 8592 |
. . . . 5
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51 | 2cnd 8651 |
. . . . 5
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52 | 2ap0 8671 |
. . . . . 6
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53 | 52 | a1i 9 |
. . . . 5
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54 | 48, 50, 51, 53 | divmulap3d 8446 |
. . . 4
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55 | 50, 51 | mulcld 7658 |
. . . . 5
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56 | 46, 47, 55 | addlsub 7999 |
. . . 4
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57 | 54, 56 | bitrd 187 |
. . 3
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58 | 44, 57 | syl5rbbr 194 |
. 2
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59 | 2, 1, 14, 43, 58 | en3i 6595 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-xor 1322 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-po 4156 df-iso 4157 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-en 6565 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-2 8637 df-n0 8830 df-z 8907 df-dvds 11289 |
This theorem is referenced by: xpnnen 11699 unennn 11702 |
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