MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sucelon Unicode version

Theorem sucelon 4608
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 4605 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
2 sucexb 4600 . . 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 4403 . 2  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
5 elon2 4403 . 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 1684   _Vcvv 2788   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  onsucmin  4612  tfindsg2  4652  oaordi  6544  oalimcl  6558  omlimcl  6576  omeulem1  6580  oeordsuc  6592  infensuc  7039  cantnflem1b  7388  cantnflem1  7391  r1ordg  7450  alephnbtwn  7698  cfsuc  7883  alephsuc3  8202  alephreg  8204  nobndlem1  24346  nobndlem8  24353  nofulllem4  24359  nofulllem5  24360  vtarsu  25886  tartarmap  25888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
  Copyright terms: Public domain W3C validator