Theorem trss 3945

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 2150	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2		sseq1 3047	. . . . 5 |- ( x = B -> ( x C_ A <-> B C_ A ) )
3	1, 2	imbi12d 232	. . . 4 |- ( x = B -> ( ( x e. A -> x C_ A ) <-> ( B e. A -> B C_ A ) ) )
4	3	imbi2d 228	. . 3 |- ( x = B -> ( ( Tr A -> ( x e. A -> x C_ A ) ) <-> ( Tr A -> ( B e. A -> B C_ A ) ) ) )
5		dftr3 3940	. . . 4 |- ( Tr A <-> A. x e. A x C_ A )
6		rsp 2423	. . . 4 |- ( A. x e. A x C_ A -> ( x e. A -> x C_ A ) )
7	5, 6	sylbi 119	. . 3 |- ( Tr A -> ( x e. A -> x C_ A ) )
8	4, 7	vtoclg 2679	. 2 |- ( B e. A -> ( Tr A -> ( B e. A -> B C_ A ) ) )
9	8	pm2.43b 51	1 |- ( Tr A -> ( B e. A -> B C_ A ) )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   A.wral 2359   C_ wss 2999   Tr wtr 3936

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937

This theorem is referenced by:  trin  3946  triun  3949  trintssm  3952  tz7.2  4181  ordelss  4206  trsucss  4250  ordsucss  4321