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Theorem grothtsk 8666
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 8655 . . . . 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 2919 . . . . . . . . 9  |-  x  e. 
_V
3 eltskg 8581 . . . . . . . . 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 8 . . . . . . . 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 676 . . . . . . 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 940 . . . . . . 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 244 . . . . . 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 1589 . . . . 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 201 . . . 4  |-  E. x
( w  e.  x  /\  x  e.  Tarski )
10 eluni 3978 . . . 4  |-  ( w  e.  U. Tarski  <->  E. x
( w  e.  x  /\  x  e.  Tarski ) )
119, 10mpbir 201 . . 3  |-  w  e. 
U. Tarski
12 vex 2919 . . 3  |-  w  e. 
_V
1311, 122th 231 . 2  |-  ( w  e.  U. Tarski  <->  w  e.  _V )
1413eqriv 2401 1  |-  U. Tarski  =  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   class class class wbr 4172    ~~ cen 7065   Tarskictsk 8579
This theorem is referenced by:  inaprc  8667  tskmval  8670  tskmcl  8672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-groth 8654
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-tsk 8580
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