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Theorem onsucsssucr 4281
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4298. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
Assertion
Ref Expression
onsucsssucr  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( suc  A  C_  suc  B  ->  A  C_  B ) )

Proof of Theorem onsucsssucr
StepHypRef Expression
1 ordsucim 4272 . . 3  |-  ( Ord 
B  ->  Ord  suc  B
)
2 ordelsuc 4277 . . 3  |-  ( ( A  e.  On  /\  Ord  suc  B )  -> 
( A  e.  suc  B  <->  suc  A  C_  suc  B ) )
31, 2sylan2 280 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  <->  suc  A  C_  suc  B ) )
4 ordtr 4161 . . . 4  |-  ( Ord 
B  ->  Tr  B
)
5 trsucss 4206 . . . 4  |-  ( Tr  B  ->  ( A  e.  suc  B  ->  A  C_  B ) )
64, 5syl 14 . . 3  |-  ( Ord 
B  ->  ( A  e.  suc  B  ->  A  C_  B ) )
76adantl 271 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  ->  A  C_  B ) )
83, 7sylbird 168 1  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( suc  A  C_  suc  B  ->  A  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434    C_ wss 2982   Tr wtr 3895   Ord word 4145   Oncon0 4146   suc csuc 4148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-uni 3622  df-tr 3896  df-iord 4149  df-suc 4154
This theorem is referenced by:  nnsucsssuc  6156
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