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Theorem finds 4521
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 4062 . . . . . 6  |-  (/)  e.  _V
3 finds.1 . . . . . 6  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
42, 3elab 2831 . . . . 5  |-  ( (/)  e.  { x  |  ph } 
<->  ps )
51, 4mpbir 145 . . . 4  |-  (/)  e.  {
x  |  ph }
6 finds.6 . . . . . 6  |-  ( y  e.  om  ->  ( ch  ->  th ) )
7 vex 2692 . . . . . . 7  |-  y  e. 
_V
8 finds.2 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
97, 8elab 2831 . . . . . 6  |-  ( y  e.  { x  | 
ph }  <->  ch )
107sucex 4422 . . . . . . 7  |-  suc  y  e.  _V
11 finds.3 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1210, 11elab 2831 . . . . . 6  |-  ( suc  y  e.  { x  |  ph }  <->  th )
136, 9, 123imtr4g 204 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )
1413rgen 2488 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ph }  ->  suc  y  e.  { x  |  ph }
)
15 peano5 4519 . . . 4  |-  ( (
(/)  e.  { x  |  ph }  /\  A. y  e.  om  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )  ->  om  C_  { x  |  ph } )
165, 14, 15mp2an 423 . . 3  |-  om  C_  { x  |  ph }
1716sseli 3097 . 2  |-  ( A  e.  om  ->  A  e.  { x  |  ph } )
18 finds.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
1918elabg 2833 . 2  |-  ( A  e.  om  ->  ( A  e.  { x  |  ph }  <->  ta )
)
2017, 19mpbid 146 1  |-  ( A  e.  om  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417    C_ wss 3075   (/)c0 3367   suc csuc 4294   omcom 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-int 3779  df-suc 4300  df-iom 4512
This theorem is referenced by:  findes  4524  nn0suc  4525  elnn  4526  ordom  4527  nndceq0  4538  0elnn  4539  omsinds  4542  nna0r  6381  nnm0r  6382  nnsucelsuc  6394  nneneq  6758  php5  6759  php5dom  6764  fidcenumlemrk  6849  fidcenumlemr  6850  frec2uzltd  10206  frecuzrdgg  10219  seq3val  10261  seqvalcd  10262  omgadd  10579  zfz1iso  10615  ennnfonelemhom  11962  nninfalllemn  13375  nninfsellemdc  13379
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