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Theorem sucelon 6636
Description: The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
Assertion
Ref Expression
sucelon  |-  ( A  e.  On  <->  suc  A  e.  On )

Proof of Theorem sucelon
StepHypRef Expression
1 ordsuc 6633 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
2 sucexb 6628 . . 3  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2anbi12i 697 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
4 elon2 4889 . 2  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
5 elon2 4889 . 2  |-  ( suc 
A  e.  On  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
63, 4, 53bitr4i 277 1  |-  ( A  e.  On  <->  suc  A  e.  On )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1767   _Vcvv 3113   Ord word 4877   Oncon0 4878   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884
This theorem is referenced by:  onsucmin  6640  tfindsg2  6680  oaordi  7195  oalimcl  7209  omlimcl  7227  omeulem1  7231  oeordsuc  7243  infensuc  7695  cantnflem1b  8105  cantnflem1  8108  cantnflem1bOLD  8128  cantnflem1OLD  8131  r1ordg  8196  alephnbtwn  8452  cfsuc  8637  alephsuc3  8955  alephreg  8957  nobndlem1  29057  nobndlem8  29064  nofulllem4  29070  nofulllem5  29071
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