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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 7758 | . . . 4 | |
2 | elin 3254 | . . . . 5 | |
3 | 2 | biimpri 132 | . . . 4 |
4 | 1, 3 | mpan2 421 | . . 3 |
5 | inss1 3291 | . . . . 5 | |
6 | ssralv 3156 | . . . . 5 | |
7 | 5, 6 | ax-mp 5 | . . . 4 |
8 | inss2 3292 | . . . . . . . 8 | |
9 | 8 | sseli 3088 | . . . . . . 7 |
10 | 1red 7774 | . . . . . . 7 | |
11 | 9, 10 | readdcld 7788 | . . . . . 6 |
12 | elin 3254 | . . . . . . 7 | |
13 | 12 | simplbi2com 1420 | . . . . . 6 |
14 | 11, 13 | syl 14 | . . . . 5 |
15 | 14 | ralimia 2491 | . . . 4 |
16 | 7, 15 | syl 14 | . . 3 |
17 | reex 7747 | . . . . 5 | |
18 | 17 | inex2 4058 | . . . 4 |
19 | eleq2 2201 | . . . . . . 7 | |
20 | eleq2 2201 | . . . . . . . 8 | |
21 | 20 | raleqbi1dv 2632 | . . . . . . 7 |
22 | 19, 21 | anbi12d 464 | . . . . . 6 |
23 | 22 | elabg 2825 | . . . . 5 |
24 | dfnn2 8715 | . . . . . 6 | |
25 | intss1 3781 | . . . . . 6 | |
26 | 24, 25 | eqsstrid 3138 | . . . . 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 3104 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cab 2123 wral 2414 cvv 2681 cin 3065 wss 3066 cint 3766 (class class class)co 5767 cr 7612 c1 7614 caddc 7616 cn 8713 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-in 3072 df-ss 3079 df-int 3767 df-inn 8714 |
This theorem is referenced by: nnssre 8717 nnind 8729 |
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