Theorem trsucss 5508

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 5489	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		trss 4506	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3		eqimss 3484	. . . 4 |- ( B = A -> B C_ A )
4	3	a1i 11	. . 3 |- ( Tr A -> ( B = A -> B C_ A ) )
5	2, 4	jaod 382	. 2 |- ( Tr A -> ( ( B e. A \/ B = A ) -> B C_ A ) )
6	1, 5	syl5 33	1 |- ( Tr A -> ( B e. suc A -> B C_ A ) )

Syntax hints:   -> wi 4   \/ wo 370   = wceq 1444   e. wcel 1887   C_ wss 3404   Tr wtr 4497   suc csuc 5425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-v 3047  df-un 3409  df-in 3411  df-ss 3418  df-sn 3969  df-uni 4199  df-tr 4498  df-suc 5429

This theorem is referenced by:  efgmnvl  17364