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Theorem tskpwss 8390
Description: 1st axiom of a Tarski's class. The subsets of an element of a Tarski's class belong to the class. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskpwss  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )

Proof of Theorem tskpwss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltskg 8388 . . . . 5  |-  ( T  e.  Tarski  ->  ( T  e. 
Tarski 
<->  ( A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  /\  A. x  e.  ~P  T
( x  ~~  T  \/  x  e.  T
) ) ) )
21ibi 232 . . . 4  |-  ( T  e.  Tarski  ->  ( A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  /\  A. x  e.  ~P  T
( x  ~~  T  \/  x  e.  T
) ) )
32simpld 445 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y ) )
4 simpl 443 . . . 4  |-  ( ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  ->  ~P x  C_  T )
54ralimi 2631 . . 3  |-  ( A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  ->  A. x  e.  T  ~P x  C_  T )
63, 5syl 15 . 2  |-  ( T  e.  Tarski  ->  A. x  e.  T  ~P x  C_  T )
7 pweq 3641 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
87sseq1d 3218 . . 3  |-  ( x  =  A  ->  ( ~P x  C_  T  <->  ~P A  C_  T ) )
98rspccva 2896 . 2  |-  ( ( A. x  e.  T  ~P x  C_  T  /\  A  e.  T )  ->  ~P A  C_  T
)
106, 9sylan 457 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039    ~~ cen 6876   Tarskictsk 8386
This theorem is referenced by:  tsksdom  8394  tskss  8396  tsktrss  8399  inttsk  8412  tskcard  8419  subtsm  25993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-tsk 8387
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