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Theorem tsktrss 8383
Description: A transitive element of a Tarski's class is a part of the class. JFM CLASSES2 th. 8 (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsktrss  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  A  C_  T
)

Proof of Theorem tsktrss
StepHypRef Expression
1 simp2 956 . . 3  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  Tr  A
)
2 dftr4 4118 . . 3  |-  ( Tr  A  <->  A  C_  ~P A
)
31, 2sylib 188 . 2  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  A  C_  ~P A )
4 tskpwss 8374 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
543adant2 974 . 2  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  ~P A  C_  T )
63, 5sstrd 3189 1  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  A  C_  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   Tr wtr 4113   Tarskictsk 8370
This theorem is referenced by:  eltintpar  25899  inttaror  25900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-tr 4114  df-tsk 8371
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