Theorem onsucsssucr 4486

Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4504. (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 4477	. . 3 |- ( Ord B -> Ord suc B )
2		ordelsuc 4482	. . 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 4356	. . . 4 |- ( Ord B -> Tr B )
5		trsucss 4401	. . . 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 2136   C_ wss 3116   Tr wtr 4080   Ord word 4340   Oncon0 4341   suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-uni 3790  df-tr 4081  df-iord 4344  df-suc 4349

This theorem is referenced by:  nnsucsssuc  6460