Theorem onsucsssucr 4433

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4450. (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 4424	. . 3 |- ( Ord B -> Ord suc B )
2		ordelsuc 4429	. . 3 |- ( ( A e. On /\ Ord suc B ) -> ( A e. suc B <-> suc A C_ suc B ) )
3	1, 2	sylan2 284	. 2 |- ( ( A e. On /\ Ord B ) -> ( A e. suc B <-> suc A C_ suc B ) )
4		ordtr 4308	. . . 4 |- ( Ord B -> Tr B )
5		trsucss 4353	. . . 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 275	. 2 |- ( ( A e. On /\ Ord B ) -> ( A e. suc B -> A C_ B ) )
8	3, 7	sylbird 169	1 |- ( ( A e. On /\ Ord B ) -> ( suc A C_ suc B -> A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1481   C_ wss 3076   Tr wtr 4034   Ord word 4292   Oncon0 4293   suc csuc 4295

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-uni 3745  df-tr 4035  df-iord 4296  df-suc 4301

This theorem is referenced by:  nnsucsssuc  6396