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Theorem sucelon 4760
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 4757 . . 3  |-  ( Ord 
A  <->  Ord  suc  A )
2 sucexb 4752 . . 3  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2anbi12i 679 . 2  |-  ( ( Ord  A  /\  A  e.  _V )  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
4 elon2 4556 . 2  |-  ( A  e.  On  <->  ( Ord  A  /\  A  e.  _V ) )
5 elon2 4556 . 2  |-  ( suc 
A  e.  On  <->  ( Ord  suc 
A  /\  suc  A  e. 
_V ) )
63, 4, 53bitr4i 269 1  |-  ( A  e.  On  <->  suc  A  e.  On )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   _Vcvv 2920   Ord word 4544   Oncon0 4545   suc csuc 4547
This theorem is referenced by:  onsucmin  4764  tfindsg2  4804  oaordi  6752  oalimcl  6766  omlimcl  6784  omeulem1  6788  oeordsuc  6800  infensuc  7248  cantnflem1b  7602  cantnflem1  7605  r1ordg  7664  alephnbtwn  7912  cfsuc  8097  alephsuc3  8415  alephreg  8417  nobndlem1  25564  nobndlem8  25571  nofulllem4  25577  nofulllem5  25578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-tr 4267  df-eprel 4458  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551
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