Theorem onsucsssucr 4281

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4298. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)

Assertion

Ref	Expression
onsucsssucr	|- ( ( A e. On /\ Ord B ) -> ( suc A C_ suc B -> A C_ B ) )

Proof of Theorem onsucsssucr

Step	Hyp	Ref	Expression
1		ordsucim 4272	. . 3 |- ( Ord B -> Ord suc B )
2		ordelsuc 4277	. . 3 |- ( ( A e. On /\ Ord suc B ) -> ( A e. suc B <-> suc A C_ suc B ) )
3	1, 2	sylan2 280	. 2 |- ( ( A e. On /\ Ord B ) -> ( A e. suc B <-> suc A C_ suc B ) )
4		ordtr 4161	. . . 4 |- ( Ord B -> Tr B )
5		trsucss 4206	. . . 4 |- ( Tr B -> ( A e. suc B -> A C_ B ) )
6	4, 5	syl 14	. . 3 |- ( Ord B -> ( A e. suc B -> A C_ B ) )
7	6	adantl 271	. 2 |- ( ( A e. On /\ Ord B ) -> ( A e. suc B -> A C_ B ) )
8	3, 7	sylbird 168	1 |- ( ( A e. On /\ Ord B ) -> ( suc A C_ suc B -> A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1434   C_ wss 2982   Tr wtr 3895   Ord word 4145   Oncon0 4146   suc csuc 4148

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-uni 3622  df-tr 3896  df-iord 4149  df-suc 4154

This theorem is referenced by:  nnsucsssuc  6156