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Theorem tskpwss 8587
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 8585 . . . . 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 233 . . . 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 446 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  T  ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y ) )
4 simpl 444 . . . 4  |-  ( ( ~P x  C_  T  /\  E. y  e.  T  ~P x  C_  y )  ->  ~P x  C_  T )
54ralimi 2745 . . 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 3766 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
87sseq1d 3339 . . 3  |-  ( x  =  A  ->  ( ~P x  C_  T  <->  ~P A  C_  T ) )
98rspccva 3015 . 2  |-  ( ( A. x  e.  T  ~P x  C_  T  /\  A  e.  T )  ->  ~P A  C_  T
)
106, 9sylan 458 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671    C_ wss 3284   ~Pcpw 3763   class class class wbr 4176    ~~ cen 7069   Tarskictsk 8583
This theorem is referenced by:  tsksdom  8591  tskss  8593  tsktrss  8596  inttsk  8609  tskcard  8616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-tsk 8584
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