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Theorem find 4872
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 967 . 2  |-  A  C_  om
3 3simpc 957 . . . . 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 8 . . . 4  |-  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
5 df-ral 2712 . . . . . 6  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
6 alral 2766 . . . . . 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 8 . . 3  |-  ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )
10 peano5 4870 . . 3  |-  ( (
(/)  e.  A  /\  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) )  ->  om  C_  A
)
119, 10ax-mp 8 . 2  |-  om  C_  A
122, 11eqssi 3366 1  |-  A  =  om
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   (/)c0 3630   suc csuc 4585   omcom 4847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848
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