Theorem trss 4096

Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
trss	|- ( Tr A -> ( B e. A -> B C_ A ) )

Proof of Theorem trss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2233	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2		sseq1 3170	. . . . 5 |- ( x = B -> ( x C_ A <-> B C_ A ) )
3	1, 2	imbi12d 233	. . . 4 |- ( x = B -> ( ( x e. A -> x C_ A ) <-> ( B e. A -> B C_ A ) ) )
4	3	imbi2d 229	. . 3 |- ( x = B -> ( ( Tr A -> ( x e. A -> x C_ A ) ) <-> ( Tr A -> ( B e. A -> B C_ A ) ) ) )
5		dftr3 4091	. . . 4 |- ( Tr A <-> A. x e. A x C_ A )
6		rsp 2517	. . . 4 |- ( A. x e. A x C_ A -> ( x e. A -> x C_ A ) )
7	5, 6	sylbi 120	. . 3 |- ( Tr A -> ( x e. A -> x C_ A ) )
8	4, 7	vtoclg 2790	. 2 |- ( B e. A -> ( Tr A -> ( B e. A -> B C_ A ) ) )
9	8	pm2.43b 52	1 |- ( Tr A -> ( B e. A -> B C_ A ) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   A.wral 2448   C_ wss 3121   Tr wtr 4087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088

This theorem is referenced by:  trin  4097  triun  4100  trintssm  4103  tz7.2  4339  ordelss  4364  trsucss  4408  ordsucss  4488  ctinf  12385