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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 7975 |
. . . 4
![]() ![]() ![]() ![]() | |
2 | elin 3333 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 2 | biimpri 133 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 1, 3 | mpan2 425 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | inss1 3370 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | ssralv 3234 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | ax-mp 5 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | inss2 3371 |
. . . . . . . 8
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9 | 8 | sseli 3166 |
. . . . . . 7
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10 | 1red 7991 |
. . . . . . 7
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11 | 9, 10 | readdcld 8006 |
. . . . . 6
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12 | elin 3333 |
. . . . . . 7
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13 | 12 | simplbi2com 1455 |
. . . . . 6
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14 | 11, 13 | syl 14 |
. . . . 5
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15 | 14 | ralimia 2551 |
. . . 4
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16 | 7, 15 | syl 14 |
. . 3
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17 | reex 7964 |
. . . . 5
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18 | 17 | inex2 4153 |
. . . 4
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19 | eleq2 2253 |
. . . . . . 7
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20 | eleq2 2253 |
. . . . . . . 8
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21 | 20 | raleqbi1dv 2694 |
. . . . . . 7
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22 | 19, 21 | anbi12d 473 |
. . . . . 6
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23 | 22 | elabg 2898 |
. . . . 5
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24 | dfnn2 8940 |
. . . . . 6
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25 | intss1 3874 |
. . . . . 6
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26 | 24, 25 | eqsstrid 3216 |
. . . . 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 3182 |
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 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-ext 2171 ax-sep 4136 ax-cnex 7921 ax-resscn 7922 ax-1re 7924 ax-addrcl 7927 |
This theorem depends on definitions: df-bi 117 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-v 2754 df-in 3150 df-ss 3157 df-int 3860 df-inn 8939 |
This theorem is referenced by: nnssre 8942 nnind 8954 |
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