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Theorem finds 6711
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
Hypotheses
Ref Expression
finds.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
finds.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
finds.3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
finds.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
finds.5  |-  ps
finds.6  |-  ( y  e.  om  ->  ( ch  ->  th ) )
Assertion
Ref Expression
finds  |-  ( A  e.  om  ->  ta )
Distinct variable groups:    x, y    x, A    ps, x    ch, x    th, x    ta, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)    ta( y)    A( y)

Proof of Theorem finds
StepHypRef Expression
1 finds.5 . . . . 5  |-  ps
2 0ex 4567 . . . . . 6  |-  (/)  e.  _V
3 finds.1 . . . . . 6  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
42, 3elab 3232 . . . . 5  |-  ( (/)  e.  { x  |  ph } 
<->  ps )
51, 4mpbir 209 . . . 4  |-  (/)  e.  {
x  |  ph }
6 finds.6 . . . . . 6  |-  ( y  e.  om  ->  ( ch  ->  th ) )
7 vex 3098 . . . . . . 7  |-  y  e. 
_V
8 finds.2 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
97, 8elab 3232 . . . . . 6  |-  ( y  e.  { x  | 
ph }  <->  ch )
107sucex 6631 . . . . . . 7  |-  suc  y  e.  _V
11 finds.3 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1210, 11elab 3232 . . . . . 6  |-  ( suc  y  e.  { x  |  ph }  <->  th )
136, 9, 123imtr4g 270 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )
1413rgen 2803 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ph }  ->  suc  y  e.  { x  |  ph }
)
15 peano5 6708 . . . 4  |-  ( (
(/)  e.  { x  |  ph }  /\  A. y  e.  om  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )  ->  om  C_  { x  |  ph } )
165, 14, 15mp2an 672 . . 3  |-  om  C_  { x  |  ph }
1716sseli 3485 . 2  |-  ( A  e.  om  ->  A  e.  { x  |  ph } )
18 finds.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
1918elabg 3233 . 2  |-  ( A  e.  om  ->  ( A  e.  { x  |  ph }  <->  ta )
)
2017, 19mpbid 210 1  |-  ( A  e.  om  ->  ta )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383    e. wcel 1804   {cab 2428   A.wral 2793    C_ wss 3461   (/)c0 3770   suc csuc 4870   omcom 6685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-om 6686
This theorem is referenced by:  findsg  6712  findes  6715  seqomlem1  7117  nna0r  7260  nnm0r  7261  nnawordi  7272  nneob  7303  nneneq  7702  pssnn  7740  inf3lem1  8048  inf3lem2  8049  cantnfval2  8091  cantnfp1lem3  8102  cantnfval2OLD  8121  cantnfp1lem3OLD  8128  r1fin  8194  ackbij1lem14  8616  ackbij1lem16  8618  ackbij1  8621  ackbij2lem2  8623  ackbij2lem3  8624  infpssrlem4  8689  fin23lem14  8716  fin23lem34  8729  itunitc1  8803  ituniiun  8805  om2uzuzi  12042  om2uzlti  12043  om2uzrdg  12049  uzrdgxfr  12059  hashgadd  12427  mreexexd  15027  trpredmintr  29290  findfvcl  29893
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