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Theorem tsken 9121
Description: Third axiom of a Tarski class. A subset of a Tarski 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 4600 . . 3  |-  ( T  e.  Tarski  ->  ( A  e. 
~P T  <->  A  C_  T
) )
21biimpar 483 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  A  e.  ~P T )
3 eltskg 9117 . . . . 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 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
) ) )
54simprd 461 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T ) )
6 breq1 4442 . . . . 5  |-  ( x  =  A  ->  (
x  ~~  T  <->  A  ~~  T ) )
7 eleq1 2526 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7orbi12d 707 . . . 4  |-  ( x  =  A  ->  (
( x  ~~  T  \/  x  e.  T
)  <->  ( A  ~~  T  \/  A  e.  T ) ) )
98rspccva 3206 . . 3  |-  ( ( A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T )  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
105, 9sylan 469 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
112, 10syldan 468 1  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461   ~Pcpw 3999   class class class wbr 4439    ~~ cen 7506   Tarskictsk 9115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-tsk 9116
This theorem is referenced by:  tskssel  9124  inttsk  9141  r1tskina  9149  tskuni  9150
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