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Theorem predon 28836
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 28835 . 2  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  ( On 
i^i  A ) )
2 onss 6597 . . 3  |-  ( A  e.  On  ->  A  C_  On )
3 sseqin2 3710 . . 3  |-  ( A 
C_  On  <->  ( On  i^i  A )  =  A )
42, 3sylib 196 . 2  |-  ( A  e.  On  ->  ( On  i^i  A )  =  A )
51, 4eqtrd 2501 1  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    i^i cin 3468    C_ wss 3469    _E cep 4782   Oncon0 4871   Predcpred 28806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-pred 28807
This theorem is referenced by:  tfrALTlem  28925  tfr2ALT  28927  tfr3ALT  28928
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