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Mirrors > Home > ILE Home > Th. List > peano5nnnn | Unicode version |
Description: Peano's inductive postulate. This is a counterpart to peano5nni 8920 designed for real number axioms which involve natural numbers (notably, axcaucvg 7898). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nntopi.n |
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Ref | Expression |
---|---|
peano5nnnn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5881 |
. . . 4
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2 | 1 | eleq1d 2246 |
. . 3
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3 | 2 | cbvralv 2703 |
. 2
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4 | ax1re 7860 |
. . . . 5
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5 | elin 3318 |
. . . . . 6
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6 | 5 | biimpri 133 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 4, 6 | mpan2 425 |
. . . 4
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8 | inss1 3355 |
. . . . . 6
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9 | ssralv 3219 |
. . . . . 6
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10 | 8, 9 | ax-mp 5 |
. . . . 5
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11 | inss2 3356 |
. . . . . . . 8
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12 | 11 | sseli 3151 |
. . . . . . 7
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13 | axaddrcl 7863 |
. . . . . . . 8
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14 | 4, 13 | mpan2 425 |
. . . . . . 7
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15 | elin 3318 |
. . . . . . . 8
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16 | 15 | simplbi2com 1444 |
. . . . . . 7
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17 | 12, 14, 16 | 3syl 17 |
. . . . . 6
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18 | 17 | ralimia 2538 |
. . . . 5
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19 | 10, 18 | syl 14 |
. . . 4
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20 | axcnex 7857 |
. . . . . . 7
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21 | axresscn 7858 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
22 | 20, 21 | ssexi 4141 |
. . . . . 6
![]() ![]() ![]() ![]() |
23 | 22 | inex2 4138 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | eleq2 2241 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | eleq2 2241 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 25 | raleqbi1dv 2680 |
. . . . . . . 8
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27 | 24, 26 | anbi12d 473 |
. . . . . . 7
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28 | 27 | elabg 2883 |
. . . . . 6
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29 | nntopi.n |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | intss1 3859 |
. . . . . . 7
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31 | 29, 30 | eqsstrid 3201 |
. . . . . 6
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32 | 28, 31 | syl6bir 164 |
. . . . 5
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33 | 23, 32 | ax-mp 5 |
. . . 4
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34 | 7, 19, 33 | syl2an 289 |
. . 3
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35 | 34, 8 | sstrdi 3167 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 3, 35 | sylan2br 288 |
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-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 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-ral 2460 df-rex 2461 df-reu 2462 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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-eprel 4289 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-1o 6416 df-2o 6417 df-oadd 6420 df-omul 6421 df-er 6534 df-ec 6536 df-qs 6540 df-ni 7302 df-pli 7303 df-mi 7304 df-lti 7305 df-plpq 7342 df-mpq 7343 df-enq 7345 df-nqqs 7346 df-plqqs 7347 df-mqqs 7348 df-1nqqs 7349 df-rq 7350 df-ltnqqs 7351 df-enq0 7422 df-nq0 7423 df-0nq0 7424 df-plq0 7425 df-mq0 7426 df-inp 7464 df-i1p 7465 df-iplp 7466 df-enr 7724 df-nr 7725 df-plr 7726 df-0r 7729 df-1r 7730 df-c 7816 df-1 7818 df-r 7820 df-add 7821 |
This theorem is referenced by: nnindnn 7891 |
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