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Theorem predon 23604
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 23603 . 2  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  ( On 
i^i  A ) )
2 onss 4582 . . 3  |-  ( A  e.  On  ->  A  C_  On )
3 sseqin2 3388 . . 3  |-  ( A 
C_  On  <->  ( On  i^i  A )  =  A )
42, 3sylib 188 . 2  |-  ( A  e.  On  ->  ( On  i^i  A )  =  A )
51, 4eqtrd 2315 1  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    _E cep 4303   Oncon0 4392   Predcpred 23578
This theorem is referenced by:  tfrALTlem  23687  tfr2ALT  23689  tfr3ALT  23690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 23579
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