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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.
StepHypRef 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
) )
31, 2imbi12d 232 . . . 4  |-  ( x  =  B  ->  (
( x  e.  A  ->  x  C_  A )  <->  ( B  e.  A  ->  B  C_  A ) ) )
43imbi2d 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 ) )
75, 6sylbi 119 . . 3  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
84, 7vtoclg 2679 . 2  |-  ( B  e.  A  ->  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) ) )
98pm2.43b 51 1  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
Colors of variables: wff set class
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
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