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Theorem peano5 4695
 Description: The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's 5 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 hypothesis, is derived from this theorem as theorem findes 4702. (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 3312 . . . . . 6
21adantl 452 . . . . 5
3 eldifi 3311 . . . . . . . . . 10
43adantl 452 . . . . . . . . 9
5 elndif 3313 . . . . . . . . . 10
6 eleq1 2356 . . . . . . . . . . . 12
76biimpcd 215 . . . . . . . . . . 11
87necon3bd 2496 . . . . . . . . . 10
95, 8mpan9 455 . . . . . . . . 9
10 nnsuc 4689 . . . . . . . . 9
114, 9, 10syl2anc 642 . . . . . . . 8
1211adantlr 695 . . . . . . 7
1312adantr 451 . . . . . 6
14 nfra1 2606 . . . . . . . . . . 11
15 nfv 1609 . . . . . . . . . . 11
1614, 15nfan 1783 . . . . . . . . . 10
17 nfv 1609 . . . . . . . . . 10
18 rsp 2616 . . . . . . . . . . 11
19 vex 2804 . . . . . . . . . . . . . . . . . 18
2019sucid 4487 . . . . . . . . . . . . . . . . 17
21 eleq2 2357 . . . . . . . . . . . . . . . . 17
2220, 21mpbiri 224 . . . . . . . . . . . . . . . 16
23 eleq1 2356 . . . . . . . . . . . . . . . . . 18
24 peano2b 4688 . . . . . . . . . . . . . . . . . 18
2523, 24syl6bbr 254 . . . . . . . . . . . . . . . . 17
26 minel 3523 . . . . . . . . . . . . . . . . . . 19
27 neldif 3314 . . . . . . . . . . . . . . . . . . 19
2826, 27sylan2 460 . . . . . . . . . . . . . . . . . 18
2928exp32 588 . . . . . . . . . . . . . . . . 17
3025, 29syl6bi 219 . . . . . . . . . . . . . . . 16
3122, 30mpid 37 . . . . . . . . . . . . . . 15
323, 31syl5 28 . . . . . . . . . . . . . 14
3332imp3a 420 . . . . . . . . . . . . 13
34 eleq1a 2365 . . . . . . . . . . . . . 14
3534com12 27 . . . . . . . . . . . . 13
3633, 35imim12d 68 . . . . . . . . . . . 12
3736com13 74 . . . . . . . . . . 11
3818, 37sylan9 638 . . . . . . . . . 10
3916, 17, 38rexlimd 2677 . . . . . . . . 9
4039exp32 588 . . . . . . . 8
4140a1i 10 . . . . . . 7
4241imp41 576 . . . . . 6
4313, 42mpd 14 . . . . 5
442, 43mtand 640 . . . 4
4544nrexdv 2659 . . 3
46 ordom 4681 . . . . 5
47 difss 3316 . . . . 5
48 tz7.5 4429 . . . . 5
4946, 47, 48mp3an12 1267 . . . 4
5049necon1bi 2502 . . 3
5145, 50syl 15 . 2
52 ssdif0 3526 . 2
5351, 52sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   wceq 1632   wcel 1696   wne 2459  wral 2556  wrex 2557   cdif 3162   cin 3164   wss 3165  c0 3468   word 4407   csuc 4410  com 4672 This theorem is referenced by:  find  4697  finds  4698  finds2  4700  omex  7360  dfom3  7364 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673
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