MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfis3 Unicode version

Theorem tfis3 4685
Description: Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
Hypotheses
Ref Expression
tfis3.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
tfis3.2  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
tfis3.3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
Assertion
Ref Expression
tfis3  |-  ( A  e.  On  ->  ch )
Distinct variable groups:    ps, x    ph, y    ch, x    x, A    x, y
Allowed substitution hints:    ph( x)    ps( y)    ch( y)    A( y)

Proof of Theorem tfis3
StepHypRef Expression
1 tfis3.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
2 tfis3.1 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
3 tfis3.3 . . 3  |-  ( x  e.  On  ->  ( A. y  e.  x  ps  ->  ph ) )
42, 3tfis2 4684 . 2  |-  ( x  e.  On  ->  ph )
51, 4vtoclga 2883 1  |-  ( A  e.  On  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   A.wral 2577   Oncon0 4429
This theorem is referenced by:  tfisi  4686  tfinds  4687  ordtypelem7  7284  rankonidlem  7545  tcrank  7599  infxpenlem  7686  alephle  7760  dfac12lem3  7816  ttukeylem5  8185  ttukeylem6  8186  tskord  8447  grudomon  8484  aomclem6  26304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
  Copyright terms: Public domain W3C validator