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Theorem onsucsssucr 4182
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4194. (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 4174 . . 3 (Ord B → Ord suc B)
2 ordelsuc 4179 . . 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 4062 . . . 4 (Ord B → Tr B)
5 trsucss 4108 . . . 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 1374  wss 2892  Tr wtr 3826  Ord word 4046  Oncon0 4047  suc csuc 4049
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  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 2897  df-in 2899  df-ss 2906  df-sn 3354  df-uni 3553  df-tr 3827  df-iord 4050  df-suc 4055
This theorem is referenced by:  nnsucsssuc  5984
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