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Theorem tskss 9086
Description: The subsets of an element of a Tarski 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 4556 . . . 4  |-  ( A  e.  T  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 464 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 tskpwss 9080 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
43sseld 3440 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  e.  ~P A  ->  B  e.  T ) )
52, 4sylbird 235 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  C_  A  ->  B  e.  T ) )
653impia 1194 1  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  C_  A )  ->  B  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    e. wcel 1842    C_ wss 3413   ~Pcpw 3954   Tarskictsk 9076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-tsk 9077
This theorem is referenced by:  tskin  9087  tsksn  9088  tsksuc  9090  tsk0  9091  tskr1om2  9096  tskint  9113
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