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Theorem 0tsk 9179
Description: The empty set is a (transitive) Tarski class. (Contributed by FL, 30-Dec-2010.)
Assertion
Ref Expression
0tsk  |-  (/)  e.  Tarski

Proof of Theorem 0tsk
StepHypRef Expression
1 ral0 3908 . 2  |-  A. x  e.  (/)  ( ~P x  C_  (/)  /\  ~P x  e.  (/) )
2 elsni 4027 . . . . 5  |-  ( x  e.  { (/) }  ->  x  =  (/) )
3 0ex 4557 . . . . . . . 8  |-  (/)  e.  _V
43enref 7609 . . . . . . 7  |-  (/)  ~~  (/)
5 breq1 4429 . . . . . . 7  |-  ( x  =  (/)  ->  ( x 
~~  (/)  <->  (/)  ~~  (/) ) )
64, 5mpbiri 236 . . . . . 6  |-  ( x  =  (/)  ->  x  ~~  (/) )
76orcd 393 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~~  (/)  \/  x  e.  (/) ) )
82, 7syl 17 . . . 4  |-  ( x  e.  { (/) }  ->  ( x  ~~  (/)  \/  x  e.  (/) ) )
9 pw0 4150 . . . 4  |-  ~P (/)  =  { (/)
}
108, 9eleq2s 2537 . . 3  |-  ( x  e.  ~P (/)  ->  (
x  ~~  (/)  \/  x  e.  (/) ) )
1110rgen 2792 . 2  |-  A. x  e.  ~P  (/) ( x  ~~  (/) 
\/  x  e.  (/) )
12 eltsk2g 9175 . . 3  |-  ( (/)  e.  _V  ->  ( (/)  e.  Tarski  <->  ( A. x  e.  (/)  ( ~P x  C_  (/)  /\  ~P x  e.  (/) )  /\  A. x  e.  ~P  (/) ( x 
~~  (/)  \/  x  e.  (/) ) ) ) )
133, 12ax-mp 5 . 2  |-  ( (/)  e.  Tarski 
<->  ( A. x  e.  (/)  ( ~P x  C_  (/) 
/\  ~P x  e.  (/) )  /\  A. x  e. 
~P  (/) ( x  ~~  (/) 
\/  x  e.  (/) ) ) )
141, 11, 13mpbir2an 928 1  |-  (/)  e.  Tarski
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   {csn 4002   class class class wbr 4426    ~~ cen 7574   Tarskictsk 9172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-en 7578  df-tsk 9173
This theorem is referenced by:  r1tskina  9206  grutsk  9246  tskmcl  9265
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