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Theorem tskpwss 9131
Description: First axiom of a Tarski class. The subsets of an element of a Tarski 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 9129 . . . . 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 241 . . . 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 459 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y ) )
4 simpl 457 . . . 4  |-  ( ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  ->  ~P x  C_  T )
54ralimi 2857 . . 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 16 . 2  |-  ( T  e.  Tarski  ->  A. x  e.  T  ~P x  C_  T )
7 pweq 4013 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
87sseq1d 3531 . . 3  |-  ( x  =  A  ->  ( ~P x  C_  T  <->  ~P A  C_  T ) )
98rspccva 3213 . 2  |-  ( ( A. x  e.  T  ~P x  C_  T  /\  A  e.  T )  ->  ~P A  C_  T
)
106, 9sylan 471 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447    ~~ cen 7514   Tarskictsk 9127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-tsk 9128
This theorem is referenced by:  tsksdom  9135  tskss  9137  tsktrss  9140  inttsk  9153  tskcard  9160
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