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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 4525. (Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
peano5 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfom3 4514 |
. . 3
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2 | peano1 4516 |
. . . . . . . 8
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3 | elin 3264 |
. . . . . . . 8
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4 | 2, 3 | mpbiran 925 |
. . . . . . 7
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5 | 4 | biimpri 132 |
. . . . . 6
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6 | peano2 4517 |
. . . . . . . . . . . 12
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7 | 6 | adantr 274 |
. . . . . . . . . . 11
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8 | 7 | a1i 9 |
. . . . . . . . . 10
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9 | pm3.31 260 |
. . . . . . . . . 10
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10 | 8, 9 | jcad 305 |
. . . . . . . . 9
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11 | 10 | alimi 1432 |
. . . . . . . 8
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12 | df-ral 2422 |
. . . . . . . 8
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13 | elin 3264 |
. . . . . . . . . 10
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14 | elin 3264 |
. . . . . . . . . 10
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15 | 13, 14 | imbi12i 238 |
. . . . . . . . 9
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16 | 15 | albii 1447 |
. . . . . . . 8
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17 | 11, 12, 16 | 3imtr4i 200 |
. . . . . . 7
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18 | df-ral 2422 |
. . . . . . 7
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19 | 17, 18 | sylibr 133 |
. . . . . 6
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20 | 5, 19 | anim12i 336 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | omex 4515 |
. . . . . . 7
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22 | 21 | inex1 4070 |
. . . . . 6
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23 | eleq2 2204 |
. . . . . . 7
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24 | eleq2 2204 |
. . . . . . . 8
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25 | 24 | raleqbi1dv 2637 |
. . . . . . 7
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26 | 23, 25 | anbi12d 465 |
. . . . . 6
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27 | 22, 26 | elab 2832 |
. . . . 5
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28 | 20, 27 | sylibr 133 |
. . . 4
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29 | intss1 3794 |
. . . 4
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30 | 28, 29 | syl 14 |
. . 3
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31 | 1, 30 | eqsstrid 3148 |
. 2
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32 | ssid 3122 |
. . . 4
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33 | 32 | biantrur 301 |
. . 3
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34 | ssin 3303 |
. . 3
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35 | 33, 34 | bitri 183 |
. 2
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36 | 31, 35 | sylibr 133 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-suc 4301 df-iom 4513 |
This theorem is referenced by: find 4521 finds 4522 finds2 4523 indpi 7174 |
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