Theorem onsucsssucr 4493

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4511. (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 4484	. . 3 |- ( Ord B -> Ord suc B )
2		ordelsuc 4489	. . 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 4363	. . . 4 |- ( Ord B -> Tr B )
5		trsucss 4408	. . . 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 2141   C_ wss 3121   Tr wtr 4087   Ord word 4347   Oncon0 4348   suc csuc 4350

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-suc 4356

This theorem is referenced by:  nnsucsssuc  6471