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Theorem onsucsssucr 4200
 Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4212. (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 4192 . . 3 (Ord B → Ord suc B)
2 ordelsuc 4197 . . 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 4081 . . . 4 (Ord B → Tr B)
5 trsucss 4126 . . . 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 1390   ⊆ wss 2911  Tr wtr 3845  Ord word 4065  Oncon0 4066  suc csuc 4068 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-suc 4074 This theorem is referenced by:  nnsucsssuc  6010
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