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Mirrors > Home > ILE Home > Th. List > peano5nni | Unicode version |
Description: Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
peano5nni |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7969 |
. . . 4
![]() ![]() ![]() ![]() | |
2 | elin 3330 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 2 | biimpri 133 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 1, 3 | mpan2 425 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | inss1 3367 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | ssralv 3231 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | ax-mp 5 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | inss2 3368 |
. . . . . . . 8
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9 | 8 | sseli 3163 |
. . . . . . 7
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10 | 1red 7985 |
. . . . . . 7
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11 | 9, 10 | readdcld 8000 |
. . . . . 6
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12 | elin 3330 |
. . . . . . 7
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13 | 12 | simplbi2com 1454 |
. . . . . 6
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14 | 11, 13 | syl 14 |
. . . . 5
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15 | 14 | ralimia 2548 |
. . . 4
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16 | 7, 15 | syl 14 |
. . 3
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17 | reex 7958 |
. . . . 5
![]() ![]() ![]() ![]() | |
18 | 17 | inex2 4150 |
. . . 4
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19 | eleq2 2251 |
. . . . . . 7
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20 | eleq2 2251 |
. . . . . . . 8
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21 | 20 | raleqbi1dv 2691 |
. . . . . . 7
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22 | 19, 21 | anbi12d 473 |
. . . . . 6
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23 | 22 | elabg 2895 |
. . . . 5
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24 | dfnn2 8934 |
. . . . . 6
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25 | intss1 3871 |
. . . . . 6
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26 | 24, 25 | eqsstrid 3213 |
. . . . 5
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27 | 23, 26 | syl6bir 164 |
. . . 4
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28 | 18, 27 | ax-mp 5 |
. . 3
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29 | 4, 16, 28 | syl2an 289 |
. 2
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30 | 29, 5 | sstrdi 3179 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-sep 4133 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-v 2751 df-in 3147 df-ss 3154 df-int 3857 df-inn 8933 |
This theorem is referenced by: nnssre 8936 nnind 8948 |
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