Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  find Unicode version

Theorem find 4348
 Description: The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that is a set of natural numbers, zero belongs to , and given any member of the member's successor also belongs to . The conclusion is that every natural number is in . (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
find.1
Assertion
Ref Expression
find
Distinct variable group:   ,

Proof of Theorem find
StepHypRef Expression
1 find.1 . . 3
21simp1i 948 . 2
3 3simpc 938 . . . . 5
41, 3ax-mp 7 . . . 4
5 df-ral 2354 . . . . . 6
6 alral 2410 . . . . . 6
75, 6sylbi 119 . . . . 5
87anim2i 334 . . . 4
94, 8ax-mp 7 . . 3
10 peano5 4347 . . 3
119, 10ax-mp 7 . 2
122, 11eqssi 3016 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   w3a 920  wal 1283   wceq 1285   wcel 1434  wral 2349   wss 2974  c0 3258   csuc 4128  com 4339 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-iinf 4337 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-suc 4134  df-iom 4340 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator