Theorem trsucss 5515

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 5496	. 2 |- ( B e. suc A -> ( B e. A \/ B = A ) )
2		trss 4499	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
3		eqimss 3470	. . . 4 |- ( B = A -> B C_ A )
4	3	a1i 11	. . 3 |- ( Tr A -> ( B = A -> B C_ A ) )
5	2, 4	jaod 387	. 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 375   = wceq 1452   e. wcel 1904   C_ wss 3390   Tr wtr 4490   suc csuc 5432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-v 3033  df-un 3395  df-in 3397  df-ss 3404  df-sn 3960  df-uni 4191  df-tr 4491  df-suc 5436

This theorem is referenced by:  efgmnvl  17442