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Theorem grothtsk 8604
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 8593 . . . . 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 2876 . . . . . . . . 9  |-  x  e. 
_V
3 eltskg 8519 . . . . . . . . 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 675 . . . . . . 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 939 . . . . . . 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 243 . . . . . 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 1587 . . . . 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 200 . . . 4  |-  E. x
( w  e.  x  /\  x  e.  Tarski )
10 eluni 3932 . . . 4  |-  ( w  e.  U. Tarski  <->  E. x
( w  e.  x  /\  x  e.  Tarski ) )
119, 10mpbir 200 . . 3  |-  w  e. 
U. Tarski
12 vex 2876 . . 3  |-  w  e. 
_V
1311, 122th 230 . 2  |-  ( w  e.  U. Tarski  <->  w  e.  _V )
1413eqriv 2363 1  |-  U. Tarski  =  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 935   E.wex 1546    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   _Vcvv 2873    C_ wss 3238   ~Pcpw 3714   U.cuni 3929   class class class wbr 4125    ~~ cen 7003   Tarskictsk 8517
This theorem is referenced by:  inaprc  8605  tskmval  8608  tskmcl  8610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-groth 8592
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-tsk 8518
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