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Theorem tsken 9025
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 4556 . . 3  |-  ( T  e.  Tarski  ->  ( A  e. 
~P T  <->  A  C_  T
) )
21biimpar 485 . 2  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  A  e.  ~P T )
3 eltskg 9021 . . . . 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 463 . . 3  |-  ( T  e.  Tarski  ->  A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T ) )
6 breq1 4396 . . . . 5  |-  ( x  =  A  ->  (
x  ~~  T  <->  A  ~~  T ) )
7 eleq1 2523 . . . . 5  |-  ( x  =  A  ->  (
x  e.  T  <->  A  e.  T ) )
86, 7orbi12d 709 . . . 4  |-  ( x  =  A  ->  (
( x  ~~  T  \/  x  e.  T
)  <->  ( A  ~~  T  \/  A  e.  T ) ) )
98rspccva 3171 . . 3  |-  ( ( A. x  e.  ~P  T ( x  ~~  T  \/  x  e.  T )  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
105, 9sylan 471 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  ~P T )  -> 
( A  ~~  T  \/  A  e.  T
) )
112, 10syldan 470 1  |-  ( ( T  e.  Tarski  /\  A  C_  T )  ->  ( A  ~~  T  \/  A  e.  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796    C_ wss 3429   ~Pcpw 3961   class class class wbr 4393    ~~ cen 7410   Tarskictsk 9019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-tsk 9020
This theorem is referenced by:  tskssel  9028  inttsk  9045  r1tskina  9053  tskuni  9054
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