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Theorem limsuc 4820
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 4819 . . 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 4657 . . . 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  4821  limuni3  4823  peano2b  4852  rdgsucg  6672  rdgsucmptnf  6678  oesuclem  6760  oaordi  6780  omordi  6800  oeordi  6821  oelim2  6829  limenpsi  7273  r1tr  7691  r1ordg  7693  r1pwss  7699  r1val1  7701  rankdmr1  7716  rankr1bg  7718  pwwf  7722  rankr1c  7736  rankonidlem  7743  ranklim  7759  r1pwcl  7762  rankxplim3  7794  infxpenlem  7884  alephordi  7944  cflm  8119  cfslb2n  8137  alephreg  8446  r1limwun  8600  rankcf  8641  inatsk  8642
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 4692
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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