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Mirrors > Home > ILE Home > Th. List > znnen | Unicode version |
Description: The set of integers and the set of positive integers are equinumerous. Corollary 8.1.23 of [AczelRathjen], p. 75. (Contributed by NM, 31-Jul-2004.) |
Ref | Expression |
---|---|
znnen |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unrab 3406 |
. . 3
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2 | nnssz 9259 |
. . . . . 6
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3 | dfss1 3339 |
. . . . . 6
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4 | 2, 3 | mpbi 145 |
. . . . 5
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5 | dfin5 3136 |
. . . . 5
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6 | 4, 5 | eqtr3i 2200 |
. . . 4
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7 | 6 | uneq1i 3285 |
. . 3
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8 | rabid2 2653 |
. . . 4
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9 | elznn 9258 |
. . . . 5
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10 | 9 | simprbi 275 |
. . . 4
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11 | 8, 10 | mprgbir 2535 |
. . 3
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12 | 1, 7, 11 | 3eqtr4ri 2209 |
. 2
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13 | nnex 8914 |
. . . 4
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14 | 13 | enref 6759 |
. . 3
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15 | zex 9251 |
. . . . . 6
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16 | 15 | rabex 4144 |
. . . . 5
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17 | nn0ex 9171 |
. . . . 5
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18 | negeq 8140 |
. . . . . . . 8
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19 | 18 | eleq1d 2246 |
. . . . . . 7
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20 | 19 | elrab 2893 |
. . . . . 6
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21 | 20 | simprbi 275 |
. . . . 5
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22 | negeq 8140 |
. . . . . . 7
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23 | 22 | eleq1d 2246 |
. . . . . 6
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24 | nn0negz 9276 |
. . . . . 6
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25 | nn0cn 9175 |
. . . . . . . . 9
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26 | 25 | negnegd 8249 |
. . . . . . . 8
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27 | 26 | eleq1d 2246 |
. . . . . . 7
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28 | 27 | ibir 177 |
. . . . . 6
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29 | 23, 24, 28 | elrabd 2895 |
. . . . 5
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30 | elrabi 2890 |
. . . . . . . 8
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31 | 30 | adantr 276 |
. . . . . . 7
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32 | 31 | zcnd 9365 |
. . . . . 6
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33 | 25 | adantl 277 |
. . . . . 6
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34 | negcon2 8200 |
. . . . . 6
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35 | 32, 33, 34 | syl2anc 411 |
. . . . 5
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36 | 16, 17, 21, 29, 35 | en3i 6765 |
. . . 4
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37 | nn0ennn 10419 |
. . . 4
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38 | 36, 37 | entri 6780 |
. . 3
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39 | inrab2 3408 |
. . . 4
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40 | incom 3327 |
. . . 4
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41 | rabeq0 3452 |
. . . . 5
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42 | 0red 7949 |
. . . . . . . 8
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43 | simpl 109 |
. . . . . . . . 9
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44 | 43 | nnred 8921 |
. . . . . . . 8
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45 | nngt0 8933 |
. . . . . . . . 9
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46 | 45 | adantr 276 |
. . . . . . . 8
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47 | nn0ge0 9190 |
. . . . . . . . . 10
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48 | 47 | adantl 277 |
. . . . . . . . 9
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49 | 44 | le0neg1d 8464 |
. . . . . . . . 9
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50 | 48, 49 | mpbird 167 |
. . . . . . . 8
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51 | 42, 44, 42, 46, 50 | ltletrd 8370 |
. . . . . . 7
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52 | 42 | ltnrd 8059 |
. . . . . . 7
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53 | 51, 52 | pm2.65da 661 |
. . . . . 6
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54 | 53, 4 | eleq2s 2272 |
. . . . 5
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55 | 41, 54 | mprgbir 2535 |
. . . 4
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56 | 39, 40, 55 | 3eqtr3i 2206 |
. . 3
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57 | unennn 12381 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
58 | 14, 38, 56, 57 | mp3an 1337 |
. 2
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59 | 12, 58 | eqbrtri 4021 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-xor 1376 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-er 6529 df-en 6735 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-n0 9166 df-z 9243 df-q 9609 df-rp 9641 df-fl 10256 df-mod 10309 df-dvds 11779 |
This theorem is referenced by: qnnen 12415 |
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