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Theorem finds2 4875
Description: Principle of Finite Induction (inference schema), using implicit substitutions. The first three 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, 29-Nov-2002.)
Hypotheses
Ref Expression
finds2.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
finds2.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
finds2.3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
finds2.4  |-  ( ta 
->  ps )
finds2.5  |-  ( y  e.  om  ->  ( ta  ->  ( ch  ->  th ) ) )
Assertion
Ref Expression
finds2  |-  ( x  e.  om  ->  ( ta  ->  ph ) )
Distinct variable groups:    x, y, ta    ps, x    ch, x    th, x    ph, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    th( y)

Proof of Theorem finds2
StepHypRef Expression
1 finds2.4 . . . . 5  |-  ( ta 
->  ps )
2 0ex 4341 . . . . . 6  |-  (/)  e.  _V
3 finds2.1 . . . . . . 7  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
43imbi2d 309 . . . . . 6  |-  ( x  =  (/)  ->  ( ( ta  ->  ph )  <->  ( ta  ->  ps ) ) )
52, 4elab 3084 . . . . 5  |-  ( (/)  e.  { x  |  ( ta  ->  ph ) }  <-> 
( ta  ->  ps ) )
61, 5mpbir 202 . . . 4  |-  (/)  e.  {
x  |  ( ta 
->  ph ) }
7 finds2.5 . . . . . . 7  |-  ( y  e.  om  ->  ( ta  ->  ( ch  ->  th ) ) )
87a2d 25 . . . . . 6  |-  ( y  e.  om  ->  (
( ta  ->  ch )  ->  ( ta  ->  th ) ) )
9 vex 2961 . . . . . . 7  |-  y  e. 
_V
10 finds2.2 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1110imbi2d 309 . . . . . . 7  |-  ( x  =  y  ->  (
( ta  ->  ph )  <->  ( ta  ->  ch )
) )
129, 11elab 3084 . . . . . 6  |-  ( y  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ch ) )
139sucex 4793 . . . . . . 7  |-  suc  y  e.  _V
14 finds2.3 . . . . . . . 8  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
1514imbi2d 309 . . . . . . 7  |-  ( x  =  suc  y  -> 
( ( ta  ->  ph )  <->  ( ta  ->  th ) ) )
1613, 15elab 3084 . . . . . 6  |-  ( suc  y  e.  { x  |  ( ta  ->  ph ) }  <->  ( ta  ->  th ) )
178, 12, 163imtr4g 263 . . . . 5  |-  ( y  e.  om  ->  (
y  e.  { x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta  ->  ph ) } ) )
1817rgen 2773 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta 
->  ph ) } )
19 peano5 4870 . . . 4  |-  ( (
(/)  e.  { x  |  ( ta  ->  ph ) }  /\  A. y  e.  om  (
y  e.  { x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta  ->  ph ) } ) )  ->  om  C_  { x  |  ( ta  ->  ph ) } )
206, 18, 19mp2an 655 . . 3  |-  om  C_  { x  |  ( ta  ->  ph ) }
2120sseli 3346 . 2  |-  ( x  e.  om  ->  x  e.  { x  |  ( ta  ->  ph ) } )
22 abid 2426 . 2  |-  ( x  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ph )
)
2321, 22sylib 190 1  |-  ( x  e.  om  ->  ( ta  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707    C_ wss 3322   (/)c0 3630   suc csuc 4585   omcom 4847
This theorem is referenced by:  finds1  4876  onnseq  6608  nnacl  6856  nnmcl  6857  nnecl  6858  nnacom  6862  nnaass  6867  nndi  6868  nnmass  6869  nnmsucr  6870  nnmcom  6871  nnmordi  6876  omsmolem  6898  isinf  7324  unblem2  7362  fiint  7385  dffi3  7438  card2inf  7525  cantnfle  7628  cantnflt  7629  cantnflem1  7647  cnfcom  7659  trcl  7666  fseqenlem1  7907  infpssrlem3  8187  fin23lem26  8207  axdc3lem2  8333  axdc4lem  8337  axdclem2  8402  wunr1om  8596  wuncval2  8624  tskr1om  8644  grothomex  8706  peano5nni  10005  neibastop2lem  26391
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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