MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  find Unicode version

Theorem find 4681
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  A is a set of natural numbers, zero belongs to 
A, and given any member of  A the member's successor also belongs to  A. The conclusion is that every natural number is in  A. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
find.1  |-  ( A 
C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
Assertion
Ref Expression
find  |-  A  =  om
Distinct variable group:    x, A

Proof of Theorem find
StepHypRef Expression
1 find.1 . . 3  |-  ( A 
C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
21simp1i 966 . 2  |-  A  C_  om
3 3simpc 956 . . . . 5  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A ) )
41, 3ax-mp 10 . . . 4  |-  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
5 df-ral 2550 . . . . . 6  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
6 alral 2603 . . . . . 6  |-  ( A. x ( x  e.  A  ->  suc  x  e.  A )  ->  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )
75, 6sylbi 189 . . . . 5  |-  ( A. x  e.  A  suc  x  e.  A  ->  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) )
87anim2i 554 . . . 4  |-  ( (
(/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A
) ) )
94, 8ax-mp 10 . . 3  |-  ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )
10 peano5 4679 . . 3  |-  ( (
(/)  e.  A  /\  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) )  ->  om  C_  A
)
119, 10ax-mp 10 . 2  |-  om  C_  A
122, 11eqssi 3197 1  |-  A  =  om
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 936   A.wal 1528    = wceq 1624    e. wcel 1685   A.wral 2545    C_ wss 3154   (/)c0 3457   suc csuc 4394   omcom 4656
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657
  Copyright terms: Public domain W3C validator