MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskpw Unicode version

Theorem tskpw 8375
Description: 2nd axiom of a 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 8373 . . . . 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 232 . . . 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 445 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T
) )
4 simpr 447 . . . 4  |-  ( ( ~P x  C_  T  /\  ~P x  e.  T
)  ->  ~P x  e.  T )
54ralimi 2618 . . 3  |-  ( A. x  e.  T  ( ~P x  C_  T  /\  ~P x  e.  T
)  ->  A. x  e.  T  ~P x  e.  T )
63, 5syl 15 . 2  |-  ( T  e.  Tarski  ->  A. x  e.  T  ~P x  e.  T
)
7 pweq 3628 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
87eleq1d 2349 . . 3  |-  ( x  =  A  ->  ( ~P x  e.  T  <->  ~P A  e.  T ) )
98rspccva 2883 . 2  |-  ( ( A. x  e.  T  ~P x  e.  T  /\  A  e.  T
)  ->  ~P A  e.  T )
106, 9sylan 457 1  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ~P A  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023    ~~ cen 6860   Tarskictsk 8370
This theorem is referenced by:  tsksn  8382  tsksuc  8384  tskr1om  8389  inttsk  8396  tskcard  8403  tskwun  8406  grutsk1  8443  pwtsm  25889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-tsk 8371
  Copyright terms: Public domain W3C validator