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Theorem grothtsk 9259
Description: The Tarski-Grothendieck Axiom, using abbreviations. (Contributed by Mario Carneiro, 28-May-2013.)
Assertion
Ref Expression
grothtsk  |-  U. Tarski  =  _V

Proof of Theorem grothtsk
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth5 9248 . . . . 5  |-  E. x
( w  e.  x  /\  A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e. 
~P  x ( y 
~~  x  \/  y  e.  x ) )
2 vex 3090 . . . . . . . . 9  |-  x  e. 
_V
3 eltskg 9174 . . . . . . . . 9  |-  ( x  e.  _V  ->  (
x  e.  Tarski  <->  ( A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e.  ~P  x
( y  ~~  x  \/  y  e.  x
) ) ) )
42, 3ax-mp 5 . . . . . . . 8  |-  ( x  e.  Tarski 
<->  ( A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e.  ~P  x
( y  ~~  x  \/  y  e.  x
) ) )
54anbi2i 698 . . . . . . 7  |-  ( ( w  e.  x  /\  x  e.  Tarski )  <->  ( w  e.  x  /\  ( A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e. 
~P  x ( y 
~~  x  \/  y  e.  x ) ) ) )
6 3anass 986 . . . . . . 7  |-  ( ( w  e.  x  /\  A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e. 
~P  x ( y 
~~  x  \/  y  e.  x ) )  <->  ( w  e.  x  /\  ( A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e. 
~P  x ( y 
~~  x  \/  y  e.  x ) ) ) )
75, 6bitr4i 255 . . . . . 6  |-  ( ( w  e.  x  /\  x  e.  Tarski )  <->  ( w  e.  x  /\  A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e.  ~P  x
( y  ~~  x  \/  y  e.  x
) ) )
87exbii 1714 . . . . 5  |-  ( E. x ( w  e.  x  /\  x  e. 
Tarski )  <->  E. x ( w  e.  x  /\  A. y  e.  x  ( ~P y  C_  x  /\  E. z  e.  x  ~P y  C_  z )  /\  A. y  e.  ~P  x
( y  ~~  x  \/  y  e.  x
) ) )
91, 8mpbir 212 . . . 4  |-  E. x
( w  e.  x  /\  x  e.  Tarski )
10 eluni 4225 . . . 4  |-  ( w  e.  U. Tarski  <->  E. x
( w  e.  x  /\  x  e.  Tarski ) )
119, 10mpbir 212 . . 3  |-  w  e. 
U. Tarski
12 vex 3090 . . 3  |-  w  e. 
_V
1311, 122th 242 . 2  |-  ( w  e.  U. Tarski  <->  w  e.  _V )
1413eqriv 2425 1  |-  U. Tarski  =  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087    C_ wss 3442   ~Pcpw 3985   U.cuni 4222   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-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-groth 9247
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-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  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-tsk 9173
This theorem is referenced by:  inaprc  9260  tskmval  9263  tskmcl  9265
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