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Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 4517. (Contributed by NM, 18-Feb-2004.) |
Ref | Expression |
---|---|
peano5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfom3 4506 | . . 3 | |
2 | peano1 4508 | . . . . . . . 8 | |
3 | elin 3259 | . . . . . . . 8 | |
4 | 2, 3 | mpbiran 924 | . . . . . . 7 |
5 | 4 | biimpri 132 | . . . . . 6 |
6 | peano2 4509 | . . . . . . . . . . . 12 | |
7 | 6 | adantr 274 | . . . . . . . . . . 11 |
8 | 7 | a1i 9 | . . . . . . . . . 10 |
9 | pm3.31 260 | . . . . . . . . . 10 | |
10 | 8, 9 | jcad 305 | . . . . . . . . 9 |
11 | 10 | alimi 1431 | . . . . . . . 8 |
12 | df-ral 2421 | . . . . . . . 8 | |
13 | elin 3259 | . . . . . . . . . 10 | |
14 | elin 3259 | . . . . . . . . . 10 | |
15 | 13, 14 | imbi12i 238 | . . . . . . . . 9 |
16 | 15 | albii 1446 | . . . . . . . 8 |
17 | 11, 12, 16 | 3imtr4i 200 | . . . . . . 7 |
18 | df-ral 2421 | . . . . . . 7 | |
19 | 17, 18 | sylibr 133 | . . . . . 6 |
20 | 5, 19 | anim12i 336 | . . . . 5 |
21 | omex 4507 | . . . . . . 7 | |
22 | 21 | inex1 4062 | . . . . . 6 |
23 | eleq2 2203 | . . . . . . 7 | |
24 | eleq2 2203 | . . . . . . . 8 | |
25 | 24 | raleqbi1dv 2634 | . . . . . . 7 |
26 | 23, 25 | anbi12d 464 | . . . . . 6 |
27 | 22, 26 | elab 2828 | . . . . 5 |
28 | 20, 27 | sylibr 133 | . . . 4 |
29 | intss1 3786 | . . . 4 | |
30 | 28, 29 | syl 14 | . . 3 |
31 | 1, 30 | eqsstrid 3143 | . 2 |
32 | ssid 3117 | . . . 4 | |
33 | 32 | biantrur 301 | . . 3 |
34 | ssin 3298 | . . 3 | |
35 | 33, 34 | bitri 183 | . 2 |
36 | 31, 35 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wcel 1480 cab 2125 wral 2416 cin 3070 wss 3071 c0 3363 cint 3771 csuc 4287 com 4504 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 |
This theorem is referenced by: find 4513 finds 4514 finds2 4515 indpi 7150 |
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