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Theorem tskss 8396
Description: The subsets of an element of a Tarski's class belong to the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 18-Jun-2013.)
Assertion
Ref Expression
tskss  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  C_  A )  ->  B  e.  T )

Proof of Theorem tskss
StepHypRef Expression
1 elpw2g 4190 . . . 4  |-  ( A  e.  T  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 452 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 tskpwss 8390 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
43sseld 3192 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  e.  ~P A  ->  B  e.  T ) )
52, 4sylbird 226 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  C_  A  ->  B  e.  T ) )
653impia 1148 1  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  C_  A )  ->  B  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   Tarskictsk 8386
This theorem is referenced by:  tskin  8397  tsksn  8398  tsksuc  8400  tsk0  8401  tskr1om2  8406  tskint  8423  vtarsuelt  25998
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  ax-sep 4157
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-br 4040  df-tsk 8387
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