Theorem onsucsssucr 4462

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4480. (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 4453	. . 3 |- ( Ord B -> Ord suc B )
2		ordelsuc 4458	. . 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 4333	. . . 4 |- ( Ord B -> Tr B )
5		trsucss 4378	. . . 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 2125   C_ wss 3098   Tr wtr 4058   Ord word 4317   Oncon0 4318   suc csuc 4320

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-sn 3562  df-uni 3769  df-tr 4059  df-iord 4321  df-suc 4326

This theorem is referenced by:  nnsucsssuc  6428