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Theorem grothac 8332
Description: The Tarksi-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 7980). This can be put in a more conventional form via ween 7546 and dfac8 7645. 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
StepHypRef Expression
1 axgroth6 8330 . . . 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 3533 . . . . . . . . . . 11  |-  ( x  =  y  ->  ~P x  =  ~P y
)
32sseq1d 3126 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  C_  u  <->  ~P y  C_  u ) )
42eleq1d 2319 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  e.  u  <->  ~P y  e.  u ) )
53, 4anbi12d 694 . . . . . . . . 9  |-  ( x  =  y  ->  (
( ~P x  C_  u  /\  ~P x  e.  u )  <->  ( ~P y  C_  u  /\  ~P y  e.  u )
) )
65rcla4va 2819 . . . . . . . 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 447 . . . . . . 7  |-  ( ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
) )  ->  ~P y  C_  u )
8 rabss 3171 . . . . . . . 8  |-  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  <->  A. x  e.  ~P  u ( x  ~<  u  ->  x  e.  u
) )
98biimpri 199 . . . . . . 7  |-  ( A. x  e.  ~P  u
( x  ~<  u  ->  x  e.  u )  ->  { x  e. 
~P u  |  x 
~<  u }  C_  u
)
10 vex 2730 . . . . . . . . . . 11  |-  y  e. 
_V
1110canth2 6899 . . . . . . . . . 10  |-  y  ~<  ~P y
12 sdomdom 6775 . . . . . . . . . 10  |-  ( y 
~<  ~P y  ->  y  ~<_  ~P y )
1311, 12ax-mp 10 . . . . . . . . 9  |-  y  ~<_  ~P y
14 vex 2730 . . . . . . . . . 10  |-  u  e. 
_V
15 ssdomg 6793 . . . . . . . . . 10  |-  ( u  e.  _V  ->  ( ~P y  C_  u  ->  ~P y  ~<_  u )
)
1614, 15ax-mp 10 . . . . . . . . 9  |-  ( ~P y  C_  u  ->  ~P y  ~<_  u )
17 domtr 6799 . . . . . . . . 9  |-  ( ( y  ~<_  ~P y  /\  ~P y  ~<_  u )  -> 
y  ~<_  u )
1813, 16, 17sylancr 647 . . . . . . . 8  |-  ( ~P y  C_  u  ->  y  ~<_  u )
19 tskwe 7467 . . . . . . . . 9  |-  ( ( u  e.  _V  /\  { x  e.  ~P u  |  x  ~<  u }  C_  u )  ->  u  e.  dom  card )
2014, 19mpan 654 . . . . . . . 8  |-  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  ->  u  e.  dom  card )
21 numdom 7549 . . . . . . . . 9  |-  ( ( u  e.  dom  card  /\  y  ~<_  u )  -> 
y  e.  dom  card )
2221expcom 426 . . . . . . . 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 1153 . . . . 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 2023 . . . 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 10 . . 3  |-  y  e. 
dom  card
2827, 102th 232 . 2  |-  ( y  e.  dom  card  <->  y  e.  _V )
2928eqriv 2250 1  |-  dom  card  =  _V
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2509   {crab 2512   _Vcvv 2727    C_ wss 3078   ~Pcpw 3530   class class class wbr 3920   dom cdm 4580    ~<_ cdom 6747    ~< csdm 6748   cardccrd 7452
This theorem is referenced by:  axgroth3  8333
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-groth 8325
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-iota 6143  df-riota 6190  df-recs 6274  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-card 7456
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