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Theorem grothtsk 8741
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 8730 . . . . 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 2965 . . . . . . . . 9  |-  x  e. 
_V
3 eltskg 8656 . . . . . . . . 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 677 . . . . . . 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 941 . . . . . . 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 245 . . . . . 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 1593 . . . . 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 202 . . . 4  |-  E. x
( w  e.  x  /\  x  e.  Tarski )
10 eluni 4042 . . . 4  |-  ( w  e.  U. Tarski  <->  E. x
( w  e.  x  /\  x  e.  Tarski ) )
119, 10mpbir 202 . . 3  |-  w  e. 
U. Tarski
12 vex 2965 . . 3  |-  w  e. 
_V
1311, 122th 232 . 2  |-  ( w  e.  U. Tarski  <->  w  e.  _V )
1413eqriv 2439 1  |-  U. Tarski  =  _V
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712   _Vcvv 2962    C_ wss 3306   ~Pcpw 3823   U.cuni 4039   class class class wbr 4237    ~~ cen 7135   Tarskictsk 8654
This theorem is referenced by:  inaprc  8742  tskmval  8745  tskmcl  8747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-groth 8729
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-tsk 8655
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