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Theorem tskpw 9161
Description: Second axiom of a Tarski class. The powerset of an element of a Tarski class belongs to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskpw  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )

Proof of Theorem tskpw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eltsk2g 9159 . . . . 5  |-  ( T  e.  Tarski  ->  ( T  e. 
Tarski 
<->  ( A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T )  /\  A. x  e.  ~P  T
( x  ~~  T  \/  x  e.  T
) ) ) )
21ibi 241 . . . 4  |-  ( T  e.  Tarski  ->  ( A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T )  /\  A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T ) ) )
32simpld 457 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T
) )
4 simpr 459 . . . 4  |-  ( ( ~P x  C_  T  /\  ~P x  e.  T
)  ->  ~P x  e.  T )
54ralimi 2797 . . 3  |-  ( A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T
)  ->  A. x  e.  T  ~P x  e.  T )
63, 5syl 17 . 2  |-  ( T  e.  Tarski  ->  A. x  e.  T  ~P x  e.  T
)
7 pweq 3958 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
87eleq1d 2471 . . 3  |-  ( x  =  A  ->  ( ~P x  e.  T  <->  ~P A  e.  T ) )
98rspccva 3159 . 2  |-  ( ( A. x  e.  T  ~P x  e.  T  /\  A  e.  T
)  ->  ~P A  e.  T )
106, 9sylan 469 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754    C_ wss 3414   ~Pcpw 3955   class class class wbr 4395    ~~ cen 7551   Tarskictsk 9156
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-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-pow 4572
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 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-tsk 9157
This theorem is referenced by:  tsksn  9168  tsksuc  9170  tskr1om  9175  inttsk  9182  tskcard  9189  tskwun  9192  grutsk1  9229  pwinfi3  35614
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