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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 4620. (Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
peano5 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfom3 4609 |
. . 3
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2 | peano1 4611 |
. . . . . . . 8
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3 | elin 3333 |
. . . . . . . 8
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4 | 2, 3 | mpbiran 942 |
. . . . . . 7
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5 | 4 | biimpri 133 |
. . . . . 6
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6 | peano2 4612 |
. . . . . . . . . . . 12
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7 | 6 | adantr 276 |
. . . . . . . . . . 11
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8 | 7 | a1i 9 |
. . . . . . . . . 10
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9 | pm3.31 262 |
. . . . . . . . . 10
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10 | 8, 9 | jcad 307 |
. . . . . . . . 9
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11 | 10 | alimi 1466 |
. . . . . . . 8
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12 | df-ral 2473 |
. . . . . . . 8
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13 | elin 3333 |
. . . . . . . . . 10
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14 | elin 3333 |
. . . . . . . . . 10
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15 | 13, 14 | imbi12i 239 |
. . . . . . . . 9
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16 | 15 | albii 1481 |
. . . . . . . 8
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17 | 11, 12, 16 | 3imtr4i 201 |
. . . . . . 7
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18 | df-ral 2473 |
. . . . . . 7
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19 | 17, 18 | sylibr 134 |
. . . . . 6
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20 | 5, 19 | anim12i 338 |
. . . . 5
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21 | omex 4610 |
. . . . . . 7
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22 | 21 | inex1 4152 |
. . . . . 6
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23 | eleq2 2253 |
. . . . . . 7
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24 | eleq2 2253 |
. . . . . . . 8
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25 | 24 | raleqbi1dv 2694 |
. . . . . . 7
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26 | 23, 25 | anbi12d 473 |
. . . . . 6
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27 | 22, 26 | elab 2896 |
. . . . 5
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28 | 20, 27 | sylibr 134 |
. . . 4
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29 | intss1 3874 |
. . . 4
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30 | 28, 29 | syl 14 |
. . 3
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31 | 1, 30 | eqsstrid 3216 |
. 2
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32 | ssid 3190 |
. . . 4
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33 | 32 | biantrur 303 |
. . 3
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34 | ssin 3372 |
. . 3
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35 | 33, 34 | bitri 184 |
. 2
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36 | 31, 35 | sylibr 134 |
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-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 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-uni 3825 df-int 3860 df-suc 4389 df-iom 4608 |
This theorem is referenced by: find 4616 finds 4617 finds2 4618 indpi 7370 |
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