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Theorem limsuc 4640
Description: The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
limsuc  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )

Proof of Theorem limsuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dflim4 4639 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
2 suceq 4457 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
32eleq1d 2349 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
43rspccv 2881 . . . 4  |-  ( A. x  e.  A  suc  x  e.  A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
543ad2ant3 978 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  ( B  e.  A  ->  suc  B  e.  A ) )
61, 5sylbi 187 . 2  |-  ( Lim 
A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
7 limord 4451 . . 3  |-  ( Lim 
A  ->  Ord  A )
8 ordtr 4406 . . 3  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 4476 . . . 4  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 423 . . 3  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
117, 8, 103syl 18 . 2  |-  ( Lim 
A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
126, 11impbid 183 1  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   (/)c0 3455   Tr wtr 4113   Ord word 4391   Lim wlim 4393   suc csuc 4394
This theorem is referenced by:  limsssuc  4641  limuni3  4643  peano2b  4672  rdgsucg  6436  rdgsucmptnf  6442  oesuclem  6524  oaordi  6544  omordi  6564  oeordi  6585  oelim2  6593  limenpsi  7036  r1tr  7448  r1ordg  7450  r1pwss  7456  r1val1  7458  rankdmr1  7473  rankr1bg  7475  pwwf  7479  rankr1c  7493  rankonidlem  7500  ranklim  7516  r1pwcl  7519  rankxplim3  7551  infxpenlem  7641  alephordi  7701  cflm  7876  cfslb2n  7894  alephreg  8204  r1limwun  8358  rankcf  8399  inatsk  8400
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-pw 3627  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-lim 4397  df-suc 4398
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