Theorem trsucss 4518

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 4498	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		trss 4194	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3		eqimss 3279	. . . 4 |- ( B = A -> B C_ A )
4	3	a1i 9	. . 3 |- ( Tr A -> ( B = A -> B C_ A ) )
5	2, 4	jaod 722	. 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 713   = wceq 1395   e. wcel 2200   C_ wss 3198   Tr wtr 4185   suc csuc 4460

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-uni 3892  df-tr 4186  df-suc 4466

This theorem is referenced by:  onsucsssucr  4605  ordpwsucss  4663  nnnninfeq  7318  bj-el2oss1o  16306  nnsf  16543