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Theorem limsuc 6657
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 6656 . . 3  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
2 suceq 4932 . . . . . 6  |-  ( x  =  B  ->  suc  x  =  suc  B )
32eleq1d 2523 . . . . 5  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
43rspccv 3204 . . . 4  |-  ( A. x  e.  A  suc  x  e.  A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
543ad2ant3 1017 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  ( B  e.  A  ->  suc  B  e.  A ) )
61, 5sylbi 195 . 2  |-  ( Lim 
A  ->  ( B  e.  A  ->  suc  B  e.  A ) )
7 limord 4926 . . 3  |-  ( Lim 
A  ->  Ord  A )
8 ordtr 4881 . . 3  |-  ( Ord 
A  ->  Tr  A
)
9 trsuc 4951 . . . 4  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
109ex 432 . . 3  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
117, 8, 103syl 20 . 2  |-  ( Lim 
A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
126, 11impbid 191 1  |-  ( Lim 
A  ->  ( B  e.  A  <->  suc  B  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   (/)c0 3783   Tr wtr 4532   Ord word 4866   Lim wlim 4868   suc csuc 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873
This theorem is referenced by:  limsssuc  6658  limuni3  6660  peano2b  6689  rdgsucg  7081  rdgsucmptnf  7087  oesuclem  7167  oaordi  7187  omordi  7207  oeordi  7228  oelim2  7236  limenpsi  7685  r1tr  8185  r1ordg  8187  r1pwss  8193  r1val1  8195  rankdmr1  8210  rankr1bg  8212  pwwf  8216  rankr1c  8230  rankonidlem  8237  ranklim  8253  r1pwcl  8256  rankxplim3  8290  infxpenlem  8382  alephordi  8446  cflm  8621  cfslb2n  8639  alephreg  8948  r1limwun  9103  rankcf  9144  inatsk  9145
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