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

Theorem tfis3 4837
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 4836 . 2  |-  ( x  e.  On  ->  ph )
51, 4vtoclga 3017 1  |-  ( A  e.  On  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   A.wral 2705   Oncon0 4581
This theorem is referenced by:  tfisi  4838  tfinds  4839  ordtypelem7  7493  rankonidlem  7754  tcrank  7808  infxpenlem  7895  alephle  7969  dfac12lem3  8025  ttukeylem5  8393  ttukeylem6  8394  tskord  8655  grudomon  8692  aomclem6  27134
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
  Copyright terms: Public domain W3C validator