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Theorem ordsssuc 4908
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 4832 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordsseleq 4851 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2sylan 471 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  ( A  e.  B  \/  A  =  B ) ) )
4 elsucg 4889 . . 3  |-  ( A  e.  On  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
54adantr 465 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
63, 5bitr4d 256 1  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  C_  B  <->  A  e.  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3431   Ord word 4821   Oncon0 4822   suc csuc 4824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-tr 4489  df-eprel 4735  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828
This theorem is referenced by:  onsssuc  4909  ordunisssuc  4924  ordpwsuc  6531  ordsucun  6541  cantnflt  7986  cantnflem1  8003  cantnfltOLD  8016  cantnflem1OLD  8026  nobndlem2  27973
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