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Theorem trss 3892
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 2142 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
2 sseq1 3021 . . . . 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 3887 . . . 4  |-  ( Tr  A  <->  A. x  e.  A  x  C_  A )
6 rsp 2412 . . . 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 2659 . 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 1285    e. wcel 1434   A.wral 2349    C_ wss 2974   Tr wtr 3883
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884
This theorem is referenced by:  trin  3893  triun  3896  trintssm  3899  tz7.2  4117  ordelss  4142  trsucss  4186  ordsucss  4256
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