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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 7733 | . . . 4 | |
2 | elin 3229 | . . . . 5 | |
3 | 2 | biimpri 132 | . . . 4 |
4 | 1, 3 | mpan2 421 | . . 3 |
5 | inss1 3266 | . . . . 5 | |
6 | ssralv 3131 | . . . . 5 | |
7 | 5, 6 | ax-mp 5 | . . . 4 |
8 | inss2 3267 | . . . . . . . 8 | |
9 | 8 | sseli 3063 | . . . . . . 7 |
10 | 1red 7749 | . . . . . . 7 | |
11 | 9, 10 | readdcld 7763 | . . . . . 6 |
12 | elin 3229 | . . . . . . 7 | |
13 | 12 | simplbi2com 1405 | . . . . . 6 |
14 | 11, 13 | syl 14 | . . . . 5 |
15 | 14 | ralimia 2470 | . . . 4 |
16 | 7, 15 | syl 14 | . . 3 |
17 | reex 7722 | . . . . 5 | |
18 | 17 | inex2 4033 | . . . 4 |
19 | eleq2 2181 | . . . . . . 7 | |
20 | eleq2 2181 | . . . . . . . 8 | |
21 | 20 | raleqbi1dv 2611 | . . . . . . 7 |
22 | 19, 21 | anbi12d 464 | . . . . . 6 |
23 | 22 | elabg 2803 | . . . . 5 |
24 | dfnn2 8690 | . . . . . 6 | |
25 | intss1 3756 | . . . . . 6 | |
26 | 24, 25 | eqsstrid 3113 | . . . . 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 3079 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cab 2103 wral 2393 cvv 2660 cin 3040 wss 3041 cint 3741 (class class class)co 5742 cr 7587 c1 7589 caddc 7591 cn 8688 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-in 3047 df-ss 3054 df-int 3742 df-inn 8689 |
This theorem is referenced by: nnssre 8692 nnind 8704 |
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