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Theorem predon 29946
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 29945 . 2  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  ( On 
i^i  A ) )
2 onss 6564 . . 3  |-  ( A  e.  On  ->  A  C_  On )
3 sseqin2 3657 . . 3  |-  ( A 
C_  On  <->  ( On  i^i  A )  =  A )
42, 3sylib 196 . 2  |-  ( A  e.  On  ->  ( On  i^i  A )  =  A )
51, 4eqtrd 2443 1  |-  ( A  e.  On  ->  Pred (  _E  ,  On ,  A
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413    _E cep 4731   Oncon0 4821   Predcpred 29916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-xp 4948  df-rel 4949  df-cnv 4950  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-pred 29917
This theorem is referenced by:  tfrALTlem  30035  tfr2ALT  30037  tfr3ALT  30038
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