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Theorem tfinds3 4803
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 2753 . . 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 4798 1  |-  ( A  e.  On  ->  ( et  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   A.wral 2666   (/)c0 3588   Oncon0 4541   Lim wlim 4542   suc csuc 4543
This theorem is referenced by:  oacl  6738  omcl  6739  oecl  6740  oawordri  6752  oaass  6763  oarec  6764  omordi  6768  omwordri  6774  odi  6781  omass  6782  oen0  6788  oewordri  6794  oeworde  6795  oeoelem  6800  omabs  6849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547
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