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Theorem grothac 8638
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8282). This can be put in a more conventional form via ween 7849 and dfac8 7948. Note that the mere existence of strongly inaccessible cardinals doesn't imply AC, but rather the particular form of the Tarski-Grothendieck axiom (see http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html). (Contributed by Mario Carneiro, 19-Apr-2013.)
Assertion
Ref Expression
grothac  |-  dom  card  =  _V

Proof of Theorem grothac
Dummy variables  x  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axgroth6 8636 . . . 4  |-  E. u
( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
)  /\  A. x  e.  ~P  u ( x 
~<  u  ->  x  e.  u ) )
2 pweq 3745 . . . . . . . . . . 11  |-  ( x  =  y  ->  ~P x  =  ~P y
)
32sseq1d 3318 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  C_  u  <->  ~P y  C_  u ) )
42eleq1d 2453 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  e.  u  <->  ~P y  e.  u ) )
53, 4anbi12d 692 . . . . . . . . 9  |-  ( x  =  y  ->  (
( ~P x  C_  u  /\  ~P x  e.  u )  <->  ( ~P y  C_  u  /\  ~P y  e.  u )
) )
65rspcva 2993 . . . . . . . 8  |-  ( ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
) )  ->  ( ~P y  C_  u  /\  ~P y  e.  u
) )
76simpld 446 . . . . . . 7  |-  ( ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
) )  ->  ~P y  C_  u )
8 rabss 3363 . . . . . . . 8  |-  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  <->  A. x  e.  ~P  u ( x  ~<  u  ->  x  e.  u
) )
98biimpri 198 . . . . . . 7  |-  ( A. x  e.  ~P  u
( x  ~<  u  ->  x  e.  u )  ->  { x  e. 
~P u  |  x 
~<  u }  C_  u
)
10 vex 2902 . . . . . . . . . . 11  |-  y  e. 
_V
1110canth2 7196 . . . . . . . . . 10  |-  y  ~<  ~P y
12 sdomdom 7071 . . . . . . . . . 10  |-  ( y 
~<  ~P y  ->  y  ~<_  ~P y )
1311, 12ax-mp 8 . . . . . . . . 9  |-  y  ~<_  ~P y
14 vex 2902 . . . . . . . . . 10  |-  u  e. 
_V
15 ssdomg 7089 . . . . . . . . . 10  |-  ( u  e.  _V  ->  ( ~P y  C_  u  ->  ~P y  ~<_  u )
)
1614, 15ax-mp 8 . . . . . . . . 9  |-  ( ~P y  C_  u  ->  ~P y  ~<_  u )
17 domtr 7096 . . . . . . . . 9  |-  ( ( y  ~<_  ~P y  /\  ~P y  ~<_  u )  -> 
y  ~<_  u )
1813, 16, 17sylancr 645 . . . . . . . 8  |-  ( ~P y  C_  u  ->  y  ~<_  u )
19 tskwe 7770 . . . . . . . . 9  |-  ( ( u  e.  _V  /\  { x  e.  ~P u  |  x  ~<  u }  C_  u )  ->  u  e.  dom  card )
2014, 19mpan 652 . . . . . . . 8  |-  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  ->  u  e.  dom  card )
21 numdom 7852 . . . . . . . . 9  |-  ( ( u  e.  dom  card  /\  y  ~<_  u )  -> 
y  e.  dom  card )
2221expcom 425 . . . . . . . 8  |-  ( y  ~<_  u  ->  ( u  e.  dom  card  ->  y  e. 
dom  card ) )
2318, 20, 22syl2im 36 . . . . . . 7  |-  ( ~P y  C_  u  ->  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  ->  y  e.  dom  card )
)
247, 9, 23syl2im 36 . . . . . 6  |-  ( ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
) )  ->  ( A. x  e.  ~P  u ( x  ~<  u  ->  x  e.  u
)  ->  y  e.  dom  card ) )
25243impia 1150 . . . . 5  |-  ( ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
)  /\  A. x  e.  ~P  u ( x 
~<  u  ->  x  e.  u ) )  -> 
y  e.  dom  card )
2625exlimiv 1641 . . . 4  |-  ( E. u ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u )  /\  A. x  e.  ~P  u ( x  ~<  u  ->  x  e.  u
) )  ->  y  e.  dom  card )
271, 26ax-mp 8 . . 3  |-  y  e. 
dom  card
2827, 102th 231 . 2  |-  ( y  e.  dom  card  <->  y  e.  _V )
2928eqriv 2384 1  |-  dom  card  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   class class class wbr 4153   dom cdm 4818    ~<_ cdom 7043    ~< csdm 7044   cardccrd 7755
This theorem is referenced by:  axgroth3  8639
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-groth 8631
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-riota 6485  df-recs 6569  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-card 7759
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