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Theorem tfinds3 4836
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 308 . 2  |-  ( x  =  (/)  ->  ( ( et  ->  ph )  <->  ( et  ->  ps ) ) )
3 tfinds3.2 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
43imbi2d 308 . 2  |-  ( x  =  y  ->  (
( et  ->  ph )  <->  ( et  ->  ch )
) )
5 tfinds3.3 . . 3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
65imbi2d 308 . 2  |-  ( x  =  suc  y  -> 
( ( et  ->  ph )  <->  ( et  ->  th ) ) )
7 tfinds3.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
87imbi2d 308 . 2  |-  ( x  =  A  ->  (
( et  ->  ph )  <->  ( et  ->  ta )
) )
9 tfinds3.5 . 2  |-  ( et 
->  ps )
10 tfinds3.6 . . 3  |-  ( y  e.  On  ->  ( et  ->  ( ch  ->  th ) ) )
1110a2d 24 . 2  |-  ( y  e.  On  ->  (
( et  ->  ch )  ->  ( et  ->  th ) ) )
12 r19.21v 2785 . . 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 24 . . 3  |-  ( Lim  x  ->  ( ( et  ->  A. y  e.  x  ch )  ->  ( et 
->  ph ) ) )
1512, 14syl5bi 209 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( et  ->  ch )  -> 
( et  ->  ph )
) )
162, 4, 6, 8, 9, 11, 15tfinds 4831 1  |-  ( A  e.  On  ->  ( et  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   A.wral 2697   (/)c0 3620   Oncon0 4573   Lim wlim 4574   suc csuc 4575
This theorem is referenced by:  oacl  6771  omcl  6772  oecl  6773  oawordri  6785  oaass  6796  oarec  6797  omordi  6801  omwordri  6807  odi  6814  omass  6815  oen0  6821  oewordri  6827  oeworde  6828  oeoelem  6833  omabs  6882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579
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