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Theorem tskss 8634
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 4364 . . . 4  |-  ( A  e.  T  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 454 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 tskpwss 8628 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
43sseld 3348 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  e.  ~P A  ->  B  e.  T ) )
52, 4sylbird 228 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  C_  A  ->  B  e.  T ) )
653impia 1151 1  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  C_  A )  ->  B  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726    C_ wss 3321   ~Pcpw 3800   Tarskictsk 8624
This theorem is referenced by:  tskin  8635  tsksn  8636  tsksuc  8638  tsk0  8639  tskr1om2  8644  tskint  8661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-tsk 8625
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