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Theorem find 6720
 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 1005 . 2
3 3simpc 995 . . . . 5
41, 3ax-mp 5 . . . 4
5 df-ral 2822 . . . . . 6
6 alral 2832 . . . . . 6
75, 6sylbi 195 . . . . 5
87anim2i 569 . . . 4
94, 8ax-mp 5 . . 3
10 peano5 6718 . . 3
119, 10ax-mp 5 . 2
122, 11eqssi 3525 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973  wal 1377   wceq 1379   wcel 1767  wral 2817   wss 3481  c0 3790   csuc 4886  com 6695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-om 6696 This theorem is referenced by: (None)
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