Theorem trsucss 4160

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 4140	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		trss 3863	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3		eqimss 2997	. . . 4 |- ( B = A -> B C_ A )
4	3	a1i 9	. . 3 |- ( Tr A -> ( B = A -> B C_ A ) )
5	2, 4	jaod 637	. 2 |- ( Tr A -> ( ( B e. A \/ B = A ) -> B C_ A ) )
6	1, 5	syl5 28	1 |- ( Tr A -> ( B e. suc A -> B C_ A ) )

Syntax hints:   -> wi 4   \/ wo 629   = wceq 1243   e. wcel 1393   C_ wss 2917   Tr wtr 3854   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-suc 4108

This theorem is referenced by:  onsucsssucr  4235  ordpwsucss  4291