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Theorem trss 4406
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 2503 . . . . 5  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
2 sseq1 3389 . . . . 5  |-  ( x  =  B  ->  (
x  C_  A  <->  B  C_  A
) )
31, 2imbi12d 320 . . . 4  |-  ( x  =  B  ->  (
( x  e.  A  ->  x  C_  A )  <->  ( B  e.  A  ->  B  C_  A ) ) )
43imbi2d 316 . . 3  |-  ( x  =  B  ->  (
( Tr  A  -> 
( x  e.  A  ->  x  C_  A )
)  <->  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A )
) ) )
5 dftr3 4401 . . . 4  |-  ( Tr  A  <->  A. x  e.  A  x  C_  A )
6 rsp 2788 . . . 4  |-  ( A. x  e.  A  x  C_  A  ->  ( x  e.  A  ->  x  C_  A ) )
75, 6sylbi 195 . . 3  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
84, 7vtoclg 3042 . 2  |-  ( B  e.  A  ->  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) ) )
98pm2.43b 50 1  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2727    C_ wss 3340   Tr wtr 4397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-v 2986  df-in 3347  df-ss 3354  df-uni 4104  df-tr 4398
This theorem is referenced by:  trin  4407  triun  4410  trint0  4414  tz7.2  4716  ordelss  4747  ordelord  4753  tz7.7  4757  trsucss  4816  tc2  7974  tcel  7977  r1ord3g  7998  r1ord2  8000  r1pwss  8003  rankwflemb  8012  r1elwf  8015  r1elssi  8024  uniwf  8038  itunitc1  8601  wunelss  8887  tskr1om2  8947  tskuni  8962  tskurn  8968  gruelss  8973  dfon2lem6  27613  dfon2lem9  27616  omsinds  27692  setindtr  29385  dford3lem1  29387  ordelordALT  31256  trsspwALT  31564  trsspwALT2  31565  trsspwALT3  31566  pwtrVD  31572  ordelordALTVD  31615
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