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Mirrors > Home > ILE Home > Th. List > peano2nnnn | Unicode version |
Description: A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 7926 designed for real number axioms which involve to natural numbers (notably, axcaucvg 6974). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
peano1nnnn.n |
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Ref | Expression |
---|---|
peano2nnnn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1nnnn.n |
. . . . . 6
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2 | 1 | eleq2i 2104 |
. . . . 5
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3 | elintg 3623 |
. . . . 5
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4 | 2, 3 | syl5bb 181 |
. . . 4
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5 | 4 | ibi 165 |
. . 3
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6 | vex 2560 |
. . . . . . . 8
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7 | eleq2 2101 |
. . . . . . . . 9
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8 | eleq2 2101 |
. . . . . . . . . 10
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9 | 8 | raleqbi1dv 2513 |
. . . . . . . . 9
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10 | 7, 9 | anbi12d 442 |
. . . . . . . 8
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11 | 6, 10 | elab 2687 |
. . . . . . 7
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12 | 11 | simprbi 260 |
. . . . . 6
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13 | oveq1 5519 |
. . . . . . . 8
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14 | 13 | eleq1d 2106 |
. . . . . . 7
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15 | 14 | rspcva 2654 |
. . . . . 6
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16 | 12, 15 | sylan2 270 |
. . . . 5
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17 | 16 | expcom 109 |
. . . 4
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18 | 17 | ralimia 2382 |
. . 3
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19 | 5, 18 | syl 14 |
. 2
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20 | df-1 6897 |
. . . . 5
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21 | 1sr 6836 |
. . . . . 6
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22 | 0r 6835 |
. . . . . 6
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23 | opexg 3964 |
. . . . . 6
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24 | 21, 22, 23 | mp2an 402 |
. . . . 5
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25 | 20, 24 | eqeltri 2110 |
. . . 4
![]() ![]() ![]() ![]() |
26 | addvalex 6920 |
. . . 4
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27 | 25, 26 | mpan2 401 |
. . 3
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28 | 1 | eleq2i 2104 |
. . . 4
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29 | elintg 3623 |
. . . 4
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30 | 28, 29 | syl5bb 181 |
. . 3
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31 | 27, 30 | syl 14 |
. 2
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32 | 19, 31 | mpbird 156 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-enr 6811 df-nr 6812 df-0r 6816 df-1r 6817 df-c 6895 df-1 6897 df-add 6900 |
This theorem is referenced by: nnindnn 6967 |
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