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Theorem onsucsssucr 4395
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4412. (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 4386 . . 3  |-  ( Ord 
B  ->  Ord  suc  B
)
2 ordelsuc 4391 . . 3  |-  ( ( A  e.  On  /\  Ord  suc  B )  -> 
( A  e.  suc  B  <->  suc  A  C_  suc  B ) )
31, 2sylan2 284 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  <->  suc  A  C_  suc  B ) )
4 ordtr 4270 . . . 4  |-  ( Ord 
B  ->  Tr  B
)
5 trsucss 4315 . . . 4  |-  ( Tr  B  ->  ( A  e.  suc  B  ->  A  C_  B ) )
64, 5syl 14 . . 3  |-  ( Ord 
B  ->  ( A  e.  suc  B  ->  A  C_  B ) )
76adantl 275 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  ->  A  C_  B ) )
83, 7sylbird 169 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 103    <-> wb 104    e. wcel 1465    C_ wss 3041   Tr wtr 3996   Ord word 4254   Oncon0 4255   suc csuc 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-uni 3707  df-tr 3997  df-iord 4258  df-suc 4263
This theorem is referenced by:  nnsucsssuc  6356
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