Theorem trsucss 4526

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 4506	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		trss 4201	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3		eqimss 3282	. . . 4 |- ( B = A -> B C_ A )
4	3	a1i 9	. . 3 |- ( Tr A -> ( B = A -> B C_ A ) )
5	2, 4	jaod 725	. 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 716   = wceq 1398   e. wcel 2202   C_ wss 3201   Tr wtr 4192   suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-uni 3899  df-tr 4193  df-suc 4474

This theorem is referenced by:  onsucsssucr  4613  ordpwsucss  4671  nnnninfeq  7387  bj-el2oss1o  16492  nnsf  16731