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Theorem sucelon 4711
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 4708 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
2 sucexb 4703 . . 3  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2anbi12i 678 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
4 elon2 4506 . 2  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
5 elon2 4506 . 2  |-  ( suc 
A  e.  On  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
63, 4, 53bitr4i 268 1  |-  ( A  e.  On  <->  suc  A  e.  On )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1715   _Vcvv 2873   Ord word 4494   Oncon0 4495   suc csuc 4497
This theorem is referenced by:  onsucmin  4715  tfindsg2  4755  oaordi  6686  oalimcl  6700  omlimcl  6718  omeulem1  6722  oeordsuc  6734  infensuc  7182  cantnflem1b  7535  cantnflem1  7538  r1ordg  7597  alephnbtwn  7845  cfsuc  8030  alephsuc3  8349  alephreg  8351  nobndlem1  25172  nobndlem8  25179  nofulllem4  25185  nofulllem5  25186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-suc 4501
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