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Theorem ordsssuc 4608
Description: A subset of an ordinal belongs to its successor. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordsssuc  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B ) )

Proof of Theorem ordsssuc
StepHypRef Expression
1 eloni 4532 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordsseleq 4551 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2sylan 458 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
4 elsucg 4589 . . 3  |-  ( A  e.  On  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
54adantr 452 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
63, 5bitr4d 248 1  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   Ord word 4521   Oncon0 4522   suc csuc 4524
This theorem is referenced by:  onsssuc  4609  ordunisssuc  4624  ordpwsuc  4735  ordsucun  4745  cantnflt  7560  cantnflem1  7578  nobndlem2  25371
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-suc 4528
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