MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfinds3 Structured version   Unicode version

Theorem tfinds3 4847
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 309 . 2  |-  ( x  =  (/)  ->  ( ( et  ->  ph )  <->  ( et  ->  ps ) ) )
3 tfinds3.2 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ch ) )
43imbi2d 309 . 2  |-  ( x  =  y  ->  (
( et  ->  ph )  <->  ( et  ->  ch )
) )
5 tfinds3.3 . . 3  |-  ( x  =  suc  y  -> 
( ph  <->  th ) )
65imbi2d 309 . 2  |-  ( x  =  suc  y  -> 
( ( et  ->  ph )  <->  ( et  ->  th ) ) )
7 tfinds3.4 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ta ) )
87imbi2d 309 . 2  |-  ( x  =  A  ->  (
( et  ->  ph )  <->  ( et  ->  ta )
) )
9 tfinds3.5 . 2  |-  ( et 
->  ps )
10 tfinds3.6 . . 3  |-  ( y  e.  On  ->  ( et  ->  ( ch  ->  th ) ) )
1110a2d 25 . 2  |-  ( y  e.  On  ->  (
( et  ->  ch )  ->  ( et  ->  th ) ) )
12 r19.21v 2795 . . 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 25 . . 3  |-  ( Lim  x  ->  ( ( et  ->  A. y  e.  x  ch )  ->  ( et 
->  ph ) ) )
1512, 14syl5bi 210 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( et  ->  ch )  -> 
( et  ->  ph )
) )
162, 4, 6, 8, 9, 11, 15tfinds 4842 1  |-  ( A  e.  On  ->  ( et  ->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2707   (/)c0 3630   Oncon0 4584   Lim wlim 4585   suc csuc 4586
This theorem is referenced by:  oacl  6782  omcl  6783  oecl  6784  oawordri  6796  oaass  6807  oarec  6808  omordi  6812  omwordri  6818  odi  6825  omass  6826  oen0  6832  oewordri  6838  oeworde  6839  oeoelem  6844  omabs  6893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590
  Copyright terms: Public domain W3C validator