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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 7689 |
. . . 4
![]() ![]() ![]() ![]() | |
2 | elin 3225 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 2 | biimpri 132 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 1, 3 | mpan2 419 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | inss1 3262 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | ssralv 3127 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | ax-mp 7 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | inss2 3263 |
. . . . . . . 8
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9 | 8 | sseli 3059 |
. . . . . . 7
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10 | 1red 7705 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 9, 10 | readdcld 7719 |
. . . . . 6
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12 | elin 3225 |
. . . . . . 7
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13 | 12 | simplbi2com 1403 |
. . . . . 6
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14 | 11, 13 | syl 14 |
. . . . 5
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15 | 14 | ralimia 2467 |
. . . 4
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16 | 7, 15 | syl 14 |
. . 3
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17 | reex 7678 |
. . . . 5
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18 | 17 | inex2 4023 |
. . . 4
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19 | eleq2 2178 |
. . . . . . 7
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20 | eleq2 2178 |
. . . . . . . 8
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21 | 20 | raleqbi1dv 2608 |
. . . . . . 7
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22 | 19, 21 | anbi12d 462 |
. . . . . 6
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23 | 22 | elabg 2799 |
. . . . 5
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24 | dfnn2 8632 |
. . . . . 6
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25 | intss1 3752 |
. . . . . 6
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26 | 24, 25 | eqsstrid 3109 |
. . . . 5
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27 | 23, 26 | syl6bir 163 |
. . . 4
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28 | 18, 27 | ax-mp 7 |
. . 3
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29 | 4, 16, 28 | syl2an 285 |
. 2
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30 | 29, 5 | syl6ss 3075 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-cnex 7636 ax-resscn 7637 ax-1re 7639 ax-addrcl 7642 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-v 2659 df-in 3043 df-ss 3050 df-int 3738 df-inn 8631 |
This theorem is referenced by: nnssre 8634 nnind 8646 |
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