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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 4269. (Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
peano5 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfom3 4258 |
. . 3
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2 | peano1 4260 |
. . . . . . . 8
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3 | elin 3120 |
. . . . . . . 8
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4 | 2, 3 | mpbiran 846 |
. . . . . . 7
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5 | 4 | biimpri 124 |
. . . . . 6
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6 | peano2 4261 |
. . . . . . . . . . . 12
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7 | 6 | adantr 261 |
. . . . . . . . . . 11
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8 | 7 | a1i 9 |
. . . . . . . . . 10
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9 | pm3.31 249 |
. . . . . . . . . 10
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10 | 8, 9 | jcad 291 |
. . . . . . . . 9
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11 | 10 | alimi 1341 |
. . . . . . . 8
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12 | df-ral 2305 |
. . . . . . . 8
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13 | elin 3120 |
. . . . . . . . . 10
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14 | elin 3120 |
. . . . . . . . . 10
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15 | 13, 14 | imbi12i 228 |
. . . . . . . . 9
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16 | 15 | albii 1356 |
. . . . . . . 8
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17 | 11, 12, 16 | 3imtr4i 190 |
. . . . . . 7
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18 | df-ral 2305 |
. . . . . . 7
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19 | 17, 18 | sylibr 137 |
. . . . . 6
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20 | 5, 19 | anim12i 321 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | omex 4259 |
. . . . . . 7
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22 | 21 | inex1 3882 |
. . . . . 6
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23 | eleq2 2098 |
. . . . . . 7
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24 | eleq2 2098 |
. . . . . . . 8
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25 | 24 | raleqbi1dv 2507 |
. . . . . . 7
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26 | 23, 25 | anbi12d 442 |
. . . . . 6
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27 | 22, 26 | elab 2681 |
. . . . 5
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28 | 20, 27 | sylibr 137 |
. . . 4
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29 | intss1 3621 |
. . . 4
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30 | 28, 29 | syl 14 |
. . 3
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31 | 1, 30 | syl5eqss 2983 |
. 2
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32 | ssid 2958 |
. . . 4
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33 | 32 | biantrur 287 |
. . 3
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34 | ssin 3153 |
. . 3
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35 | 33, 34 | bitri 173 |
. 2
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36 | 31, 35 | sylibr 137 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 df-suc 4074 df-iom 4257 |
This theorem is referenced by: find 4265 finds 4266 finds2 4267 indpi 6326 |
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