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Theorem predon 25217
Description: For an ordinal, the predecessor under  _E and  On is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)
Assertion
Ref Expression
predon  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  A )

Proof of Theorem predon
StepHypRef Expression
1 predep 25216 . 2  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  ( On 
i^i  A ) )
2 onss 4711 . . 3  |-  ( A  e.  On  ->  A  C_  On )
3 sseqin2 3503 . . 3  |-  ( A 
C_  On  <->  ( On  i^i  A )  =  A )
42, 3sylib 189 . 2  |-  ( A  e.  On  ->  ( On  i^i  A )  =  A )
51, 4eqtrd 2419 1  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    i^i cin 3262    C_ wss 3263    _E cep 4433   Oncon0 4522   Predcpred 25191
This theorem is referenced by:  tfrALTlem  25300  tfr2ALT  25302  tfr3ALT  25303
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-xp 4824  df-rel 4825  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-pred 25192
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