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Theorem limsuc 4821
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 4820 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
2 suceq 4638 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
32eleq1d 2501 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
43rspccv 3041 . . . 4  |-  ( A. x  e.  A  suc  x  e.  A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
543ad2ant3 980 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  ( B  e.  A  ->  suc  B  e.  A ) )
61, 5sylbi 188 . 2  |-  ( Lim 
A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
7 limord 4632 . . 3  |-  ( Lim 
A  ->  Ord  A )
8 ordtr 4587 . . 3  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 4658 . . . 4  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 424 . . 3  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
117, 8, 103syl 19 . 2  |-  ( Lim 
A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
126, 11impbid 184 1  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   (/)c0 3620   Tr wtr 4294   Ord word 4572   Lim wlim 4574   suc csuc 4575
This theorem is referenced by:  limsssuc  4822  limuni3  4824  peano2b  4853  rdgsucg  6673  rdgsucmptnf  6679  oesuclem  6761  oaordi  6781  omordi  6801  oeordi  6822  oelim2  6830  limenpsi  7274  r1tr  7692  r1ordg  7694  r1pwss  7700  r1val1  7702  rankdmr1  7717  rankr1bg  7719  pwwf  7723  rankr1c  7737  rankonidlem  7744  ranklim  7760  r1pwcl  7763  rankxplim3  7795  infxpenlem  7885  alephordi  7945  cflm  8120  cfslb2n  8138  alephreg  8447  r1limwun  8601  rankcf  8642  inatsk  8643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579
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