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Theorem onsucsssucr 4316
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4333. (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 4307 . . 3  |-  ( Ord 
B  ->  Ord  suc  B
)
2 ordelsuc 4312 . . 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 4196 . . . 4  |-  ( Ord 
B  ->  Tr  B
)
5 trsucss 4241 . . . 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 1438    C_ wss 2997   Tr wtr 3928   Ord word 4180   Oncon0 4181   suc csuc 4183
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189
This theorem is referenced by:  nnsucsssuc  6235
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