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Theorem tsktrss 8636
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 958 . . 3  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  Tr  A
)
2 dftr4 4307 . . 3  |-  ( Tr  A  <->  A  C_  ~P A
)
31, 2sylib 189 . 2  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  A  C_  ~P A )
4 tskpwss 8627 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
543adant2 976 . 2  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  ~P A  C_  T )
63, 5sstrd 3358 1  |-  ( ( T  e.  Tarski  /\  Tr  A  /\  A  e.  T
)  ->  A  C_  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    e. wcel 1725    C_ wss 3320   ~Pcpw 3799   Tr wtr 4302   Tarskictsk 8623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-tr 4303  df-tsk 8624
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