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Theorem tskss 8380
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 4174 . . . 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 8374 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
43sseld 3179 . . 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 1684    C_ wss 3152   ~Pcpw 3625   Tarskictsk 8370
This theorem is referenced by:  tskin  8381  tsksn  8382  tsksuc  8384  tsk0  8385  tskr1om2  8390  tskint  8407  vtarsuelt  25895
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  ax-sep 4141
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-br 4024  df-tsk 8371
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