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Theorem predon 27673
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 27672 . 2  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  ( On 
i^i  A ) )
2 onss 6421 . . 3  |-  ( A  e.  On  ->  A  C_  On )
3 sseqin2 3588 . . 3  |-  ( A 
C_  On  <->  ( On  i^i  A )  =  A )
42, 3sylib 196 . 2  |-  ( A  e.  On  ->  ( On  i^i  A )  =  A )
51, 4eqtrd 2475 1  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    i^i cin 3346    C_ wss 3347    _E cep 4649   Oncon0 4738   Predcpred 27643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-tr 4405  df-eprel 4651  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-xp 4865  df-rel 4866  df-cnv 4867  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 27644
This theorem is referenced by:  tfrALTlem  27762  tfr2ALT  27764  tfr3ALT  27765
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