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Theorem tfinds3 4655
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.)
Hypotheses
Ref Expression
tfinds3.1  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
tfinds3.2  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
tfinds3.3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
tfinds3.4  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
tfinds3.5  |-  ( et 
->  ps )
tfinds3.6  |-  ( y  e.  On  ->  ( et  ->  ( ch  ->  th ) ) )
tfinds3.7  |-  ( Lim  x  ->  ( et  ->  ( A. y  e.  x  ch  ->  ph )
) )
Assertion
Ref Expression
tfinds3  |-  ( A  e.  On  ->  ( et  ->  ta ) )
Distinct variable groups:    x, A    ph, y    ch, x    ta, x    x, y, et
Allowed substitution hints:    ph( x)    ps( x, y)    ch( y)    th( x, y)    ta( y)    A( y)

Proof of Theorem tfinds3
StepHypRef Expression
1 tfinds3.1 . . 3  |-  ( x  =  (/)  ->  ( ph  <->  ps ) )
21imbi2d 307 . 2  |-  ( x  =  (/)  ->  ( ( et  ->  ph )  <->  ( et  ->  ps ) ) )
3 tfinds3.2 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
43imbi2d 307 . 2  |-  ( x  =  y  ->  (
( et  ->  ph )  <->  ( et  ->  ch )
) )
5 tfinds3.3 . . 3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
65imbi2d 307 . 2  |-  ( x  =  suc  y  -> 
( ( et  ->  ph )  <->  ( et  ->  th ) ) )
7 tfinds3.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
87imbi2d 307 . 2  |-  ( x  =  A  ->  (
( et  ->  ph )  <->  ( et  ->  ta )
) )
9 tfinds3.5 . 2  |-  ( et 
->  ps )
10 tfinds3.6 . . 3  |-  ( y  e.  On  ->  ( et  ->  ( ch  ->  th ) ) )
1110a2d 23 . 2  |-  ( y  e.  On  ->  (
( et  ->  ch )  ->  ( et  ->  th ) ) )
12 r19.21v 2630 . . 3  |-  ( A. y  e.  x  ( et  ->  ch )  <->  ( et  ->  A. y  e.  x  ch ) )
13 tfinds3.7 . . . 4  |-  ( Lim  x  ->  ( et  ->  ( A. y  e.  x  ch  ->  ph )
) )
1413a2d 23 . . 3  |-  ( Lim  x  ->  ( ( et  ->  A. y  e.  x  ch )  ->  ( et 
->  ph ) ) )
1512, 14syl5bi 208 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( et  ->  ch )  -> 
( et  ->  ph )
) )
162, 4, 6, 8, 9, 11, 15tfinds 4650 1  |-  ( A  e.  On  ->  ( et  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   (/)c0 3455   Oncon0 4392   Lim wlim 4393   suc csuc 4394
This theorem is referenced by:  oacl  6534  omcl  6535  oecl  6536  oawordri  6548  oaass  6559  oarec  6560  omordi  6564  omwordri  6570  odi  6577  omass  6578  oen0  6584  oewordri  6590  oeworde  6591  oeoelem  6596  omabs  6645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398
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