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Theorem finds 4619
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 hypothesis. Theorem Schema 22 of [Suppes] p. 136. (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 4090 . . . . . 6  |-  (/)  e.  _V
3 finds.1 . . . . . 6  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
42, 3elab 2865 . . . . 5  |-  ( (/)  e.  { x  |  ph } 
<->  ps )
51, 4mpbir 202 . . . 4  |-  (/)  e.  {
x  |  ph }
6 finds.6 . . . . . 6  |-  ( y  e.  om  ->  ( ch  ->  th ) )
7 vex 2743 . . . . . . 7  |-  y  e. 
_V
8 finds.2 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
97, 8elab 2865 . . . . . 6  |-  ( y  e.  { x  | 
ph }  <->  ch )
107sucex 4539 . . . . . . 7  |-  suc  y  e.  _V
11 finds.3 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1210, 11elab 2865 . . . . . 6  |-  ( suc  y  e.  { x  |  ph }  <->  th )
136, 9, 123imtr4g 263 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )
1413rgen 2579 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ph }  ->  suc  y  e.  { x  |  ph }
)
15 peano5 4616 . . . 4  |-  ( (
(/)  e.  { x  |  ph }  /\  A. y  e.  om  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )  ->  om  C_  { x  |  ph } )
165, 14, 15mp2an 656 . . 3  |-  om  C_  { x  |  ph }
1716sseli 3118 . 2  |-  ( A  e.  om  ->  A  e.  { x  |  ph } )
18 finds.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
1918elabg 2866 . 2  |-  ( A  e.  om  ->  ( A  e.  { x  |  ph }  <->  ta )
)
2017, 19mpbid 203 1  |-  ( A  e.  om  ->  ta )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   {cab 2242   A.wral 2516    C_ wss 3094   (/)c0 3397   suc csuc 4331   omcom 4593
This theorem is referenced by:  findsg  4620  findes  4623  seqomlem1  6395  nna0r  6540  nnm0r  6541  nnawordi  6552  nneob  6583  nneneq  6977  pssnn  7014  inf3lem1  7262  inf3lem2  7263  cantnfval2  7303  cantnfp1lem3  7315  r1fin  7378  ackbij1lem14  7792  ackbij1lem16  7794  ackbij1  7797  ackbij2lem2  7799  ackbij2lem3  7800  infpssrlem4  7865  fin23lem14  7892  fin23lem34  7905  itunitc1  7979  ituniiun  7981  om2uzuzi  10943  om2uzlti  10944  om2uzrdg  10950  uzrdgxfr  10960  hashgadd  11290  mreexexd  13477  trpredmintr  23568  findfvcl  24231
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594
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