Theorem trss 4112

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 2240	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2		sseq1 3180	. . . . 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 4107	. . . 4 |- ( Tr A <-> A. x e. A x C_ A )
6		rsp 2524	. . . 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 2799	. 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 1353   e. wcel 2148   A.wral 2455   C_ wss 3131   Tr wtr 4103

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-tr 4104

This theorem is referenced by:  trin  4113  triun  4116  trintssm  4119  tz7.2  4356  ordelss  4381  trsucss  4425  ordsucss  4505  ctinf  12433