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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 3421 |
. . 3
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2 | nnssz 9301 |
. . . . . 6
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3 | dfss1 3354 |
. . . . . 6
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4 | 2, 3 | mpbi 145 |
. . . . 5
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5 | dfin5 3151 |
. . . . 5
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6 | 4, 5 | eqtr3i 2212 |
. . . 4
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7 | 6 | uneq1i 3300 |
. . 3
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8 | rabid2 2667 |
. . . 4
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9 | elznn 9300 |
. . . . 5
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10 | 9 | simprbi 275 |
. . . 4
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11 | 8, 10 | mprgbir 2548 |
. . 3
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12 | 1, 7, 11 | 3eqtr4ri 2221 |
. 2
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13 | nnex 8956 |
. . . 4
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14 | 13 | enref 6792 |
. . 3
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15 | zex 9293 |
. . . . . 6
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16 | 15 | rabex 4162 |
. . . . 5
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17 | nn0ex 9213 |
. . . . 5
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18 | negeq 8181 |
. . . . . . . 8
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19 | 18 | eleq1d 2258 |
. . . . . . 7
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20 | 19 | elrab 2908 |
. . . . . 6
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21 | 20 | simprbi 275 |
. . . . 5
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22 | negeq 8181 |
. . . . . . 7
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23 | 22 | eleq1d 2258 |
. . . . . 6
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24 | nn0negz 9318 |
. . . . . 6
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25 | nn0cn 9217 |
. . . . . . . . 9
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26 | 25 | negnegd 8290 |
. . . . . . . 8
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27 | 26 | eleq1d 2258 |
. . . . . . 7
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28 | 27 | ibir 177 |
. . . . . 6
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29 | 23, 24, 28 | elrabd 2910 |
. . . . 5
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30 | elrabi 2905 |
. . . . . . . 8
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31 | 30 | adantr 276 |
. . . . . . 7
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32 | 31 | zcnd 9407 |
. . . . . 6
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33 | 25 | adantl 277 |
. . . . . 6
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34 | negcon2 8241 |
. . . . . 6
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35 | 32, 33, 34 | syl2anc 411 |
. . . . 5
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36 | 16, 17, 21, 29, 35 | en3i 6798 |
. . . 4
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37 | nn0ennn 10466 |
. . . 4
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38 | 36, 37 | entri 6813 |
. . 3
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39 | inrab2 3423 |
. . . 4
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40 | incom 3342 |
. . . 4
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41 | rabeq0 3467 |
. . . . 5
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42 | 0red 7989 |
. . . . . . . 8
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43 | simpl 109 |
. . . . . . . . 9
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44 | 43 | nnred 8963 |
. . . . . . . 8
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45 | nngt0 8975 |
. . . . . . . . 9
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46 | 45 | adantr 276 |
. . . . . . . 8
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47 | nn0ge0 9232 |
. . . . . . . . . 10
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48 | 47 | adantl 277 |
. . . . . . . . 9
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49 | 44 | le0neg1d 8505 |
. . . . . . . . 9
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50 | 48, 49 | mpbird 167 |
. . . . . . . 8
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51 | 42, 44, 42, 46, 50 | ltletrd 8411 |
. . . . . . 7
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52 | 42 | ltnrd 8100 |
. . . . . . 7
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53 | 51, 52 | pm2.65da 662 |
. . . . . 6
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54 | 53, 4 | eleq2s 2284 |
. . . . 5
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55 | 41, 54 | mprgbir 2548 |
. . . 4
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56 | 39, 40, 55 | 3eqtr3i 2218 |
. . 3
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57 | unennn 12451 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
58 | 14, 38, 56, 57 | mp3an 1348 |
. 2
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59 | 12, 58 | eqbrtri 4039 |
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 ax-arch 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 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-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 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-res 4656 df-ima 4657 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-1st 6166 df-2nd 6167 df-er 6560 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-q 9652 df-rp 9686 df-fl 10303 df-mod 10356 df-dvds 11830 |
This theorem is referenced by: qnnen 12485 |
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