ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onsucsssucr Unicode version

Theorem onsucsssucr 4393
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4410. (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 4384 . . 3  |-  ( Ord 
B  ->  Ord  suc  B
)
2 ordelsuc 4389 . . 3  |-  ( ( A  e.  On  /\  Ord  suc  B )  -> 
( A  e.  suc  B  <->  suc  A  C_  suc  B ) )
31, 2sylan2 282 . 2  |-  ( ( A  e.  On  /\  Ord  B )  ->  ( A  e.  suc  B  <->  suc  A  C_  suc  B ) )
4 ordtr 4268 . . . 4  |-  ( Ord 
B  ->  Tr  B
)
5 trsucss 4313 . . . 4  |-  ( Tr  B  ->  ( A  e.  suc  B  ->  A  C_  B ) )
64, 5syl 14 . . 3  |-  ( Ord 
B  ->  ( A  e.  suc  B  ->  A  C_  B ) )
76adantl 273 . 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 1463    C_ wss 3039   Tr wtr 3994   Ord word 4252   Oncon0 4253   suc csuc 4255
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-uni 3705  df-tr 3995  df-iord 4256  df-suc 4261
This theorem is referenced by:  nnsucsssuc  6354
  Copyright terms: Public domain W3C validator