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Theorem grothtsk 8473
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 8462 . . . . 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 2804 . . . . . . . . 9  |-  x  e. 
_V
3 eltskg 8388 . . . . . . . . 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 938 . . . . . . 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 1572 . . . . 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 3846 . . . 4  |-  ( w  e.  U. Tarski  <->  E. x
( w  e.  x  /\  x  e.  Tarski ) )
119, 10mpbir 200 . . 3  |-  w  e. 
U. Tarski
12 vex 2804 . . 3  |-  w  e. 
_V
1311, 122th 230 . 2  |-  ( w  e.  U. Tarski  <->  w  e.  _V )
1413eqriv 2293 1  |-  U. Tarski  =  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039    ~~ cen 6876   Tarskictsk 8386
This theorem is referenced by:  inaprc  8474  tskmval  8477  tskmcl  8479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-groth 8461
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-tsk 8387
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