Theorem trss 4030

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 2200	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2		sseq1 3115	. . . . 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 4025	. . . 4 |- ( Tr A <-> A. x e. A x C_ A )
6		rsp 2478	. . . 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 2741	. 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 1331   e. wcel 1480   A.wral 2414   C_ wss 3066   Tr wtr 4021

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022

This theorem is referenced by:  trin  4031  triun  4034  trintssm  4037  tz7.2  4271  ordelss  4296  trsucss  4340  ordsucss  4415  ctinf  11932