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Theorem sucelon 6658
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 6655 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
2 sucexb 6650 . . 3  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2anbi12i 701 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
4 elon2 5453 . 2  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
5 elon2 5453 . 2  |-  ( suc 
A  e.  On  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
63, 4, 53bitr4i 280 1  |-  ( A  e.  On  <->  suc  A  e.  On )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    e. wcel 1872   _Vcvv 3080   Ord word 5441   Oncon0 5442   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-tr 4519  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445  df-on 5446  df-suc 5448
This theorem is referenced by:  onsucmin  6662  tfindsg2  6702  oaordi  7258  oalimcl  7272  omlimcl  7290  omeulem1  7294  oeordsuc  7306  infensuc  7759  cantnflem1b  8199  cantnflem1  8202  r1ordg  8257  alephnbtwn  8509  cfsuc  8694  alephsuc3  9012  alephreg  9014  nobndlem1  30586  nobndlem8  30593  nofulllem4  30599  nofulllem5  30600
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