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Theorem 0tsk 8922
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 3784 . 2  |-  A. x  e.  (/)  ( ~P x  C_  (/)  /\  ~P x  e.  (/) )
2 elsni 3902 . . . . 5  |-  ( x  e.  { (/) }  ->  x  =  (/) )
3 0ex 4422 . . . . . . . 8  |-  (/)  e.  _V
43enref 7342 . . . . . . 7  |-  (/)  ~~  (/)
5 breq1 4295 . . . . . . 7  |-  ( x  =  (/)  ->  ( x 
~~  (/)  <->  (/)  ~~  (/) ) )
64, 5mpbiri 233 . . . . . 6  |-  ( x  =  (/)  ->  x  ~~  (/) )
76orcd 392 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~~  (/)  \/  x  e.  (/) ) )
82, 7syl 16 . . . 4  |-  ( x  e.  { (/) }  ->  ( x  ~~  (/)  \/  x  e.  (/) ) )
9 pw0 4020 . . . 4  |-  ~P (/)  =  { (/)
}
108, 9eleq2s 2535 . . 3  |-  ( x  e.  ~P (/)  ->  (
x  ~~  (/)  \/  x  e.  (/) ) )
1110rgen 2781 . 2  |-  A. x  e.  ~P  (/) ( x  ~~  (/) 
\/  x  e.  (/) )
12 eltsk2g 8918 . . 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 911 1  |-  (/)  e.  Tarski
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   class class class wbr 4292    ~~ cen 7307   Tarskictsk 8915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-en 7311  df-tsk 8916
This theorem is referenced by:  r1tskina  8949  grutsk  8989  tskmcl  9008
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