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Theorem trss 4035
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 2202 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
2 sseq1 3120 . . . . 5  |-  ( x  =  B  ->  (
x  C_  A  <->  B  C_  A
) )
31, 2imbi12d 233 . . . 4  |-  ( x  =  B  ->  (
( x  e.  A  ->  x  C_  A )  <->  ( B  e.  A  ->  B  C_  A ) ) )
43imbi2d 229 . . 3  |-  ( x  =  B  ->  (
( Tr  A  -> 
( x  e.  A  ->  x  C_  A )
)  <->  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A )
) ) )
5 dftr3 4030 . . . 4  |-  ( Tr  A  <->  A. x  e.  A  x  C_  A )
6 rsp 2480 . . . 4  |-  ( A. x  e.  A  x  C_  A  ->  ( x  e.  A  ->  x  C_  A ) )
75, 6sylbi 120 . . 3  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
84, 7vtoclg 2746 . 2  |-  ( B  e.  A  ->  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) ) )
98pm2.43b 52 1  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   A.wral 2416    C_ wss 3071   Tr wtr 4026
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 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027
This theorem is referenced by:  trin  4036  triun  4039  trintssm  4042  tz7.2  4276  ordelss  4301  trsucss  4345  ordsucss  4420  ctinf  11954
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