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Theorem ordsssuc 5527
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 5451 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordsseleq 5470 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2sylan 478 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
4 elsucg 5508 . . 3  |-  ( A  e.  On  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
54adantr 471 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
63, 5bitr4d 264 1  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1454    e. wcel 1897    C_ wss 3415   Ord word 5440   Oncon0 5441   suc csuc 5443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-tr 4511  df-eprel 4763  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-ord 5444  df-on 5445  df-suc 5447
This theorem is referenced by:  onsssuc  5528  ordunisssuc  5543  ordpwsuc  6668  ordsucun  6678  cantnflt  8202  cantnflem1  8219  nobndlem2  30630
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