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Mirrors > Home > ILE Home > Th. List > peano5 | Unicode version |
Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 4418. (Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
peano5 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfom3 4407 |
. . 3
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2 | peano1 4409 |
. . . . . . . 8
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3 | elin 3183 |
. . . . . . . 8
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4 | 2, 3 | mpbiran 886 |
. . . . . . 7
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5 | 4 | biimpri 131 |
. . . . . 6
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6 | peano2 4410 |
. . . . . . . . . . . 12
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7 | 6 | adantr 270 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 7 | a1i 9 |
. . . . . . . . . 10
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9 | pm3.31 258 |
. . . . . . . . . 10
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10 | 8, 9 | jcad 301 |
. . . . . . . . 9
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11 | 10 | alimi 1389 |
. . . . . . . 8
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12 | df-ral 2364 |
. . . . . . . 8
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13 | elin 3183 |
. . . . . . . . . 10
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14 | elin 3183 |
. . . . . . . . . 10
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15 | 13, 14 | imbi12i 237 |
. . . . . . . . 9
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16 | 15 | albii 1404 |
. . . . . . . 8
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17 | 11, 12, 16 | 3imtr4i 199 |
. . . . . . 7
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18 | df-ral 2364 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 17, 18 | sylibr 132 |
. . . . . 6
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20 | 5, 19 | anim12i 331 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | omex 4408 |
. . . . . . 7
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22 | 21 | inex1 3973 |
. . . . . 6
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23 | eleq2 2151 |
. . . . . . 7
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24 | eleq2 2151 |
. . . . . . . 8
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25 | 24 | raleqbi1dv 2570 |
. . . . . . 7
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26 | 23, 25 | anbi12d 457 |
. . . . . 6
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27 | 22, 26 | elab 2760 |
. . . . 5
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28 | 20, 27 | sylibr 132 |
. . . 4
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29 | intss1 3703 |
. . . 4
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30 | 28, 29 | syl 14 |
. . 3
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31 | 1, 30 | syl5eqss 3070 |
. 2
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32 | ssid 3044 |
. . . 4
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33 | 32 | biantrur 297 |
. . 3
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34 | ssin 3222 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
35 | 33, 34 | bitri 182 |
. 2
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36 | 31, 35 | sylibr 132 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-suc 4198 df-iom 4406 |
This theorem is referenced by: find 4414 finds 4415 finds2 4416 indpi 6901 |
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