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Theorem grothac 9204
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8845). This can be put in a more conventional form via ween 8412 and dfac8 8511. 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.) (New usage is discouraged.)
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 9202 . . . 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 4013 . . . . . . . . . . 11  |-  ( x  =  y  ->  ~P x  =  ~P y
)
32sseq1d 3531 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  C_  u  <->  ~P y  C_  u ) )
42eleq1d 2536 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  e.  u  <->  ~P y  e.  u ) )
53, 4anbi12d 710 . . . . . . . . 9  |-  ( x  =  y  ->  (
( ~P x  C_  u  /\  ~P x  e.  u )  <->  ( ~P y  C_  u  /\  ~P y  e.  u )
) )
65rspcva 3212 . . . . . . . 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 459 . . . . . . 7  |-  ( ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
) )  ->  ~P y  C_  u )
8 rabss 3577 . . . . . . . 8  |-  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  <->  A. x  e.  ~P  u ( x  ~<  u  ->  x  e.  u
) )
98biimpri 206 . . . . . . 7  |-  ( A. x  e.  ~P  u
( x  ~<  u  ->  x  e.  u )  ->  { x  e. 
~P u  |  x 
~<  u }  C_  u
)
10 vex 3116 . . . . . . . . . . 11  |-  y  e. 
_V
1110canth2 7667 . . . . . . . . . 10  |-  y  ~<  ~P y
12 sdomdom 7540 . . . . . . . . . 10  |-  ( y 
~<  ~P y  ->  y  ~<_  ~P y )
1311, 12ax-mp 5 . . . . . . . . 9  |-  y  ~<_  ~P y
14 vex 3116 . . . . . . . . . 10  |-  u  e. 
_V
15 ssdomg 7558 . . . . . . . . . 10  |-  ( u  e.  _V  ->  ( ~P y  C_  u  ->  ~P y  ~<_  u )
)
1614, 15ax-mp 5 . . . . . . . . 9  |-  ( ~P y  C_  u  ->  ~P y  ~<_  u )
17 domtr 7565 . . . . . . . . 9  |-  ( ( y  ~<_  ~P y  /\  ~P y  ~<_  u )  -> 
y  ~<_  u )
1813, 16, 17sylancr 663 . . . . . . . 8  |-  ( ~P y  C_  u  ->  y  ~<_  u )
19 tskwe 8327 . . . . . . . . 9  |-  ( ( u  e.  _V  /\  { x  e.  ~P u  |  x  ~<  u }  C_  u )  ->  u  e.  dom  card )
2014, 19mpan 670 . . . . . . . 8  |-  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  ->  u  e.  dom  card )
21 numdom 8415 . . . . . . . . 9  |-  ( ( u  e.  dom  card  /\  y  ~<_  u )  -> 
y  e.  dom  card )
2221expcom 435 . . . . . . . 8  |-  ( y  ~<_  u  ->  ( u  e.  dom  card  ->  y  e. 
dom  card ) )
2318, 20, 22syl2im 38 . . . . . . 7  |-  ( ~P y  C_  u  ->  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  ->  y  e.  dom  card )
)
247, 9, 23syl2im 38 . . . . . 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 1193 . . . . 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 1698 . . . 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 5 . . 3  |-  y  e. 
dom  card
2827, 102th 239 . 2  |-  ( y  e.  dom  card  <->  y  e.  _V )
2928eqriv 2463 1  |-  dom  card  =  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   dom cdm 4999    ~<_ cdom 7511    ~< csdm 7512   cardccrd 8312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-groth 9197
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-recs 7039  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-card 8316
This theorem is referenced by:  axgroth3  9205
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