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Theorem onsucsssucr 4157
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4169. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
Assertion
Ref Expression
onsucsssucr ((A On Ord B) → (suc A ⊆ suc BAB))

Proof of Theorem onsucsssucr
StepHypRef Expression
1 ordsucim 4149 . . 3 (Ord B → Ord suc B)
2 ordelsuc 4154 . . 3 ((A On Ord suc B) → (A suc B ↔ suc A ⊆ suc B))
31, 2sylan2 270 . 2 ((A On Ord B) → (A suc B ↔ suc A ⊆ suc B))
4 ordtr 4038 . . . 4 (Ord B → Tr B)
5 trsucss 4083 . . . 4 (Tr B → (A suc BAB))
64, 5syl 14 . . 3 (Ord B → (A suc BAB))
76adantl 262 . 2 ((A On Ord B) → (A suc BAB))
83, 7sylbird 159 1 ((A On Ord B) → (suc A ⊆ suc BAB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1375  wss 2895  Tr wtr 3806  Ord word 4023  Oncon0 4024  suc csuc 4026
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-sn 3333  df-uni 3533  df-tr 3807  df-iord 4027  df-suc 4031
This theorem is referenced by: (None)
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