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Theorem finds2 4656
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 4124 . . . . . 6  |-  (/)  e.  _V
3 finds2.1 . . . . . . 7  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
43imbi2d 309 . . . . . 6  |-  ( x  =  (/)  ->  ( ( ta  ->  ph )  <->  ( ta  ->  ps ) ) )
52, 4elab 2889 . . . . 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 2766 . . . . . . 7  |-  y  e. 
_V
10 finds2.2 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
1110imbi2d 309 . . . . . . 7  |-  ( x  =  y  ->  (
( ta  ->  ph )  <->  ( ta  ->  ch )
) )
129, 11elab 2889 . . . . . 6  |-  ( y  e.  { x  |  ( ta  ->  ph ) } 
<->  ( ta  ->  ch ) )
139sucex 4574 . . . . . . 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 2889 . . . . . 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 2583 . . . 4  |-  A. y  e.  om  ( y  e. 
{ x  |  ( ta  ->  ph ) }  ->  suc  y  e.  { x  |  ( ta 
->  ph ) } )
19 peano5 4651 . . . 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 656 . . 3  |-  om  C_  { x  |  ( ta  ->  ph ) }
2120sseli 3151 . 2  |-  ( x  e.  om  ->  x  e.  { x  |  ( ta  ->  ph ) } )
22 abid 2246 . 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 6    <-> wb 178    = wceq 1619    e. wcel 1621   {cab 2244   A.wral 2518    C_ wss 3127   (/)c0 3430   suc csuc 4366   omcom 4628
This theorem is referenced by:  finds1  4657  onnseq  6329  nnacl  6577  nnmcl  6578  nnecl  6579  nnacom  6583  nnaass  6588  nndi  6589  nnmass  6590  nnmsucr  6591  nnmcom  6592  nnmordi  6597  omsmolem  6619  isinf  7044  unblem2  7078  fiint  7101  dffi3  7152  card2inf  7237  cantnfle  7340  cantnflt  7341  cantnflem1  7359  cnfcom  7371  trcl  7378  fseqenlem1  7619  infpssrlem3  7899  fin23lem26  7919  axdc3lem2  8045  axdc4lem  8049  axdclem2  8115  wunr1om  8309  wuncval2  8337  tskr1om  8357  grothomex  8419  peano5nni  9717  neibastop2lem  25677
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629
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