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Theorem limsuc 4864
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 4863 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
2 suceq 4681 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
32eleq1d 2509 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
43rspccv 3058 . . . 4  |-  ( A. x  e.  A  suc  x  e.  A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
543ad2ant3 981 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  ( B  e.  A  ->  suc  B  e.  A ) )
61, 5sylbi 189 . 2  |-  ( Lim 
A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
7 limord 4675 . . 3  |-  ( Lim 
A  ->  Ord  A )
8 ordtr 4630 . . 3  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 4701 . . . 4  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 425 . . 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 185 1  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1654    e. wcel 1728   A.wral 2712   (/)c0 3616   Tr wtr 4333   Ord word 4615   Lim wlim 4617   suc csuc 4618
This theorem is referenced by:  limsssuc  4865  limuni3  4867  peano2b  4896  rdgsucg  6717  rdgsucmptnf  6723  oesuclem  6805  oaordi  6825  omordi  6845  oeordi  6866  oelim2  6874  limenpsi  7318  r1tr  7738  r1ordg  7740  r1pwss  7746  r1val1  7748  rankdmr1  7763  rankr1bg  7765  pwwf  7769  rankr1c  7783  rankonidlem  7790  ranklim  7806  r1pwcl  7809  rankxplim3  7843  infxpenlem  7933  alephordi  7993  cflm  8168  cfslb2n  8186  alephreg  8495  r1limwun  8649  rankcf  8690  inatsk  8691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-tr 4334  df-eprel 4529  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622
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