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Theorem tsktrss 8399
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 4134 . . 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 8390 . . 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 3202 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 1696    C_ wss 3165   ~Pcpw 3638   Tr wtr 4129   Tarskictsk 8386
This theorem is referenced by:  eltintpar  26002  inttaror  26003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-tr 4130  df-tsk 8387
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