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Theorem onsucsssucr 4425
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4442. (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 4416 . . 3 (Ord 𝐵 → Ord suc 𝐵)
2 ordelsuc 4421 . . 3 ((𝐴 ∈ On ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
31, 2sylan2 284 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
4 ordtr 4300 . . . 4 (Ord 𝐵 → Tr 𝐵)
5 trsucss 4345 . . . 4 (Tr 𝐵 → (𝐴 ∈ suc 𝐵𝐴𝐵))
64, 5syl 14 . . 3 (Ord 𝐵 → (𝐴 ∈ suc 𝐵𝐴𝐵))
76adantl 275 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵𝐴𝐵))
83, 7sylbird 169 1 ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1480  wss 3071  Tr wtr 4026  Ord word 4284  Oncon0 4285  suc csuc 4287
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-uni 3737  df-tr 4027  df-iord 4288  df-suc 4293
This theorem is referenced by:  nnsucsssuc  6388
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