Theorem trsucss 4353

Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)

Assertion

Ref	Expression
trsucss	|- ( Tr A -> ( B e. suc A -> B C_ A ) )

Proof of Theorem trsucss

Step	Hyp	Ref	Expression
1		elsuci 4333	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		trss 4043	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3		eqimss 3156	. . . 4 |- ( B = A -> B C_ A )
4	3	a1i 9	. . 3 |- ( Tr A -> ( B = A -> B C_ A ) )
5	2, 4	jaod 707	. 2 |- ( Tr A -> ( ( B e. A \/ B = A ) -> B C_ A ) )
6	1, 5	syl5 32	1 |- ( Tr A -> ( B e. suc A -> B C_ A ) )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1332   e. wcel 1481   C_ wss 3076   Tr wtr 4034   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-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-uni 3745  df-tr 4035  df-suc 4301

This theorem is referenced by:  onsucsssucr  4433  ordpwsucss  4490  bj-el2oss1o  13152  nnsf  13374  nninfalllemn  13377