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Theorem find 4618
 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 969 . 2
3 3simpc 959 . . . . 5
41, 3ax-mp 10 . . . 4
5 df-ral 2520 . . . . . 6
6 alral 2572 . . . . . 6
75, 6sylbi 189 . . . . 5
87anim2i 555 . . . 4
94, 8ax-mp 10 . . 3
10 peano5 4616 . . 3
119, 10ax-mp 10 . 2
122, 11eqssi 3137 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   w3a 939  wal 1532   wceq 1619   wcel 1621  wral 2516   wss 3094  c0 3397   csuc 4331  com 4593 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594
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