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Theorem onsucsssucr 4172
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4184. (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 4164 . . 3 (Ord B → Ord suc B)
2 ordelsuc 4169 . . 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 4053 . . . 4 (Ord B → Tr B)
5 trsucss 4098 . . . 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 1366  wss 2885  Tr wtr 3817  Ord word 4037  Oncon0 4038  suc csuc 4040
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-uni 3544  df-tr 3818  df-iord 4041  df-suc 4046
This theorem is referenced by:  nnsucsssuc  5974
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