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Theorem onsucsssucr 4235
 Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4252. (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 4226 . . 3 (Ord 𝐵 → Ord suc 𝐵)
2 ordelsuc 4231 . . 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 4115 . . . 4 (Ord 𝐵 → Tr 𝐵)
5 trsucss 4160 . . . 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 1393   ⊆ wss 2917  Tr wtr 3854  Ord word 4099  Oncon0 4100  suc csuc 4102 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-suc 4108 This theorem is referenced by:  nnsucsssuc  6071
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