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Theorem onsucsssucr 4184
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4196. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
Assertion
Ref Expression
onsucsssucr  On  Ord  suc  C_  suc 
C_

Proof of Theorem onsucsssucr
StepHypRef Expression
1 ordsucim 4176 . . 3  Ord  Ord  suc
2 ordelsuc 4181 . . 3  On  Ord  suc  suc  suc  C_  suc
31, 2sylan2 270 . 2  On  Ord  suc  suc  C_ 
suc
4 ordtr 4064 . . . 4  Ord  Tr
5 trsucss 4110 . . . 4  Tr  suc 
C_
64, 5syl 14 . . 3  Ord  suc  C_
76adantl 262 . 2  On  Ord  suc 
C_
83, 7sylbird 159 1  On  Ord  suc  C_  suc 
C_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wcel 1374    C_ wss 2894   Tr wtr 3828   Ord word 4048   Oncon0 4049   suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-uni 3555  df-tr 3829  df-iord 4052  df-suc 4057
This theorem is referenced by:  nnsucsssuc  5986
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