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Theorem find 4572
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 969 . 2  |-  A  C_  om
3 3simpc 959 . . . . 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 2513 . . . . . 6  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
6 alral 2563 . . . . . 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 555 . . . 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 4570 . . 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 3116 1  |-  A  =  om
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   A.wal 1532    = wceq 1619    e. wcel 1621   A.wral 2509    C_ wss 3078   (/)c0 3362   suc csuc 4287   omcom 4547
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548
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