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Theorem tskss 9040
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 4566 . . . 4  |-  ( A  e.  T  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
21adantl 466 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
3 tskpwss 9034 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
43sseld 3466 . . 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 1185 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 369    /\ w3a 965    e. wcel 1758    C_ wss 3439   ~Pcpw 3971   Tarskictsk 9030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-tsk 9031
This theorem is referenced by:  tskin  9041  tsksn  9042  tsksuc  9044  tsk0  9045  tskr1om2  9050  tskint  9067
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