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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7889 | . . . 4 | |
2 | elin 3300 | . . . . 5 | |
3 | 2 | biimpri 132 | . . . 4 |
4 | 1, 3 | mpan2 422 | . . 3 |
5 | inss1 3337 | . . . . 5 | |
6 | ssralv 3201 | . . . . 5 | |
7 | 5, 6 | ax-mp 5 | . . . 4 |
8 | inss2 3338 | . . . . . . . 8 | |
9 | 8 | sseli 3133 | . . . . . . 7 |
10 | 1red 7905 | . . . . . . 7 | |
11 | 9, 10 | readdcld 7919 | . . . . . 6 |
12 | elin 3300 | . . . . . . 7 | |
13 | 12 | simplbi2com 1431 | . . . . . 6 |
14 | 11, 13 | syl 14 | . . . . 5 |
15 | 14 | ralimia 2525 | . . . 4 |
16 | 7, 15 | syl 14 | . . 3 |
17 | reex 7878 | . . . . 5 | |
18 | 17 | inex2 4111 | . . . 4 |
19 | eleq2 2228 | . . . . . . 7 | |
20 | eleq2 2228 | . . . . . . . 8 | |
21 | 20 | raleqbi1dv 2667 | . . . . . . 7 |
22 | 19, 21 | anbi12d 465 | . . . . . 6 |
23 | 22 | elabg 2867 | . . . . 5 |
24 | dfnn2 8850 | . . . . . 6 | |
25 | intss1 3833 | . . . . . 6 | |
26 | 24, 25 | eqsstrid 3183 | . . . . 5 |
27 | 23, 26 | syl6bir 163 | . . . 4 |
28 | 18, 27 | ax-mp 5 | . . 3 |
29 | 4, 16, 28 | syl2an 287 | . 2 |
30 | 29, 5 | sstrdi 3149 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 cab 2150 wral 2442 cvv 2721 cin 3110 wss 3111 cint 3818 (class class class)co 5836 cr 7743 c1 7745 caddc 7747 cn 8848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-in 3117 df-ss 3124 df-int 3819 df-inn 8849 |
This theorem is referenced by: nnssre 8852 nnind 8864 |
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