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Theorem tsken 8376
Description: 3rd axiom of a Tarski's class. A subset of a Tarski's class is either equipotent to the class or an element of the class. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tsken  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )

Proof of Theorem tsken
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpw2g 4174 . . 3  |-  ( T  e.  Tarski  ->  ( A  e. 
~P T  <->  A  C_  T
) )
21biimpar 471 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  A  e.  ~P T )
3 eltskg 8372 . . . . 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
) ) ) )
43ibi 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
) ) )
54simprd 449 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T ) )
6 breq1 4026 . . . . 5  |-  ( x  =  A  ->  (
x  ~~  T  <->  A  ~~  T ) )
7 eleq1 2343 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7orbi12d 690 . . . 4  |-  ( x  =  A  ->  (
( x  ~~  T  \/  x  e.  T
)  <->  ( A  ~~  T  \/  A  e.  T ) ) )
98rspccva 2883 . . 3  |-  ( ( A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T )  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
105, 9sylan 457 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
112, 10syldan 456 1  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  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   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023    ~~ cen 6860   Tarskictsk 8370
This theorem is referenced by:  tskssel  8379  inttsk  8396  r1tskina  8404  tskuni  8405  subtareqbe  25891  inttarcar  25901
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
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
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