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Theorem peano5 4321
 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 4326. (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 dfom3 4315 . . 3
2 peano1 4317 . . . . . . . 8
3 elin 3126 . . . . . . . 8
42, 3mpbiran 847 . . . . . . 7
54biimpri 124 . . . . . 6
6 peano2 4318 . . . . . . . . . . . 12
76adantr 261 . . . . . . . . . . 11
87a1i 9 . . . . . . . . . 10
9 pm3.31 249 . . . . . . . . . 10
108, 9jcad 291 . . . . . . . . 9
1110alimi 1344 . . . . . . . 8
12 df-ral 2311 . . . . . . . 8
13 elin 3126 . . . . . . . . . 10
14 elin 3126 . . . . . . . . . 10
1513, 14imbi12i 228 . . . . . . . . 9
1615albii 1359 . . . . . . . 8
1711, 12, 163imtr4i 190 . . . . . . 7
18 df-ral 2311 . . . . . . 7
1917, 18sylibr 137 . . . . . 6
205, 19anim12i 321 . . . . 5
21 omex 4316 . . . . . . 7
2221inex1 3891 . . . . . 6
23 eleq2 2101 . . . . . . 7
24 eleq2 2101 . . . . . . . 8
2524raleqbi1dv 2513 . . . . . . 7
2623, 25anbi12d 442 . . . . . 6
2722, 26elab 2687 . . . . 5
2820, 27sylibr 137 . . . 4
29 intss1 3630 . . . 4
3028, 29syl 14 . . 3
311, 30syl5eqss 2989 . 2
32 ssid 2964 . . . 4
3332biantrur 287 . . 3
34 ssin 3159 . . 3
3533, 34bitri 173 . 2
3631, 35sylibr 137 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97  wal 1241   wceq 1243   wcel 1393  cab 2026  wral 2306   cin 2916   wss 2917  c0 3224  cint 3615   csuc 4102  com 4313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314 This theorem is referenced by:  find  4322  finds  4323  finds2  4324  indpi  6440
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