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Theorem peano5 6632
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, except our proof does not require the Axiom of Infinity. 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 6639. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
peano5
Distinct variable group:   ,

Proof of Theorem peano5
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldifn 3593 . . . . . 6
21adantl 466 . . . . 5
3 eldifi 3592 . . . . . . . . . 10
43adantl 466 . . . . . . . . 9
5 elndif 3594 . . . . . . . . . 10
6 eleq1 2526 . . . . . . . . . . . 12
76biimpcd 224 . . . . . . . . . . 11
87necon3bd 2665 . . . . . . . . . 10
95, 8mpan9 469 . . . . . . . . 9
10 nnsuc 6626 . . . . . . . . 9
114, 9, 10syl2anc 661 . . . . . . . 8
1211adantlr 714 . . . . . . 7
1312adantr 465 . . . . . 6
14 nfra1 2811 . . . . . . . . . . 11
15 nfv 1674 . . . . . . . . . . 11
1614, 15nfan 1866 . . . . . . . . . 10
17 nfv 1674 . . . . . . . . . 10
18 rsp 2895 . . . . . . . . . . 11
19 vex 3084 . . . . . . . . . . . . . . . . . 18
2019sucid 4915 . . . . . . . . . . . . . . . . 17
21 eleq2 2527 . . . . . . . . . . . . . . . . 17
2220, 21mpbiri 233 . . . . . . . . . . . . . . . 16
23 eleq1 2526 . . . . . . . . . . . . . . . . . 18
24 peano2b 6625 . . . . . . . . . . . . . . . . . 18
2523, 24syl6bbr 263 . . . . . . . . . . . . . . . . 17
26 minel 3848 . . . . . . . . . . . . . . . . . . 19
27 neldif 3595 . . . . . . . . . . . . . . . . . . 19
2826, 27sylan2 474 . . . . . . . . . . . . . . . . . 18
2928exp32 605 . . . . . . . . . . . . . . . . 17
3025, 29syl6bi 228 . . . . . . . . . . . . . . . 16
3122, 30mpid 41 . . . . . . . . . . . . . . 15
323, 31syl5 32 . . . . . . . . . . . . . 14
3332impd 431 . . . . . . . . . . . . 13
34 eleq1a 2537 . . . . . . . . . . . . . 14
3534com12 31 . . . . . . . . . . . . 13
3633, 35imim12d 74 . . . . . . . . . . . 12
3736com13 80 . . . . . . . . . . 11
3818, 37sylan9 657 . . . . . . . . . 10
3916, 17, 38rexlimd 2947 . . . . . . . . 9
4039exp32 605 . . . . . . . 8
4140a1i 11 . . . . . . 7
4241imp41 593 . . . . . 6
4313, 42mpd 15 . . . . 5
442, 43mtand 659 . . . 4
4544nrexdv 2927 . . 3
46 ordom 6618 . . . . 5
47 difss 3597 . . . . 5
48 tz7.5 4857 . . . . 5
4946, 47, 48mp3an12 1305 . . . 4
5049necon1bi 2686 . . 3
5145, 50syl 16 . 2
52 ssdif0 3851 . 2
5351, 52sylibr 212 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1370  e.wcel 1758  =/=wne 2648  A.wral 2800  E.wrex 2801  \cdif 3439  i^icin 3441  C_wss 3442   c0 3751  Ordword 4835  succsuc 4838   com 6609
This theorem is referenced by:  find  6634  finds  6635  finds2  6637  omex  7986  dfom3  7990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4530  ax-nul 4538  ax-pr 4648  ax-un 6505
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3083  df-sbc 3298  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3752  df-if 3906  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4209  df-br 4410  df-opab 4468  df-tr 4503  df-eprel 4749  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-ord 4839  df-on 4840  df-lim 4841  df-suc 4842  df-om 6610
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