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Theorem onsssuc 4965
Description: A subset of an ordinal number belongs to its successor. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onsssuc  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )

Proof of Theorem onsssuc
StepHypRef Expression
1 eloni 4888 . 2  |-  ( B  e.  On  ->  Ord  B )
2 ordsssuc 4964 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
31, 2sylan2 474 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    C_ wss 3476   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-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
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-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:  ordsssuc2  4966  onmindif  4967  tfindsg  6673  dfom2  6680  findsg  6705  ondif2  7149  oeeui  7248  cantnflem1  8104  cantnflem1OLD  8127  rankr1bg  8217  rankr1c  8235  cofsmo  8645  cfsmolem  8646  cfcof  8650  fin1a2lem9  8784  alephreg  8953  winainflem  9067  nobndlem8  29036  onsuct0  29483  onint1  29491
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