Theorem trss 4196

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 2294	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2		sseq1 3250	. . . . 5 |- ( x = B -> ( x C_ A <-> B C_ A ) )
3	1, 2	imbi12d 234	. . . 4 |- ( x = B -> ( ( x e. A -> x C_ A ) <-> ( B e. A -> B C_ A ) ) )
4	3	imbi2d 230	. . 3 |- ( x = B -> ( ( Tr A -> ( x e. A -> x C_ A ) ) <-> ( Tr A -> ( B e. A -> B C_ A ) ) ) )
5		dftr3 4191	. . . 4 |- ( Tr A <-> A. x e. A x C_ A )
6		rsp 2579	. . . 4 |- ( A. x e. A x C_ A -> ( x e. A -> x C_ A ) )
7	5, 6	sylbi 121	. . 3 |- ( Tr A -> ( x e. A -> x C_ A ) )
8	4, 7	vtoclg 2864	. 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 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   Tr wtr 4187

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188

This theorem is referenced by:  trin  4197  triun  4200  trintssm  4203  tz7.2  4451  ordelss  4476  trsucss  4520  ordsucss  4602  ctinf  13050