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Theorem tskpw 8628
Description: 2nd axiom of a Tarski's class. The powerset of an element of a Tarski's 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 8626 . . . . 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 233 . . . 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 446 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T
) )
4 simpr 448 . . . 4  |-  ( ( ~P x  C_  T  /\  ~P x  e.  T
)  ->  ~P x  e.  T )
54ralimi 2781 . . 3  |-  ( A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T
)  ->  A. x  e.  T  ~P x  e.  T )
63, 5syl 16 . 2  |-  ( T  e.  Tarski  ->  A. x  e.  T  ~P x  e.  T
)
7 pweq 3802 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
87eleq1d 2502 . . 3  |-  ( x  =  A  ->  ( ~P x  e.  T  <->  ~P A  e.  T ) )
98rspccva 3051 . 2  |-  ( ( A. x  e.  T  ~P x  e.  T  /\  A  e.  T
)  ->  ~P A  e.  T )
106, 9sylan 458 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ~Pcpw 3799   class class class wbr 4212    ~~ cen 7106   Tarskictsk 8623
This theorem is referenced by:  tsksn  8635  tsksuc  8637  tskr1om  8642  inttsk  8649  tskcard  8656  tskwun  8659  grutsk1  8696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-pow 4377
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-tsk 8624
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