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Theorem grothac 9197
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8840). This can be put in a more conventional form via ween 8407 and dfac8 8506. 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 9195 . . . 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 4002 . . . . . . . . . . 11  |-  ( x  =  y  ->  ~P x  =  ~P y
)
32sseq1d 3516 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  C_  u  <->  ~P y  C_  u ) )
42eleq1d 2523 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  e.  u  <->  ~P y  e.  u ) )
53, 4anbi12d 708 . . . . . . . . 9  |-  ( x  =  y  ->  (
( ~P x  C_  u  /\  ~P x  e.  u )  <->  ( ~P y  C_  u  /\  ~P y  e.  u )
) )
65rspcva 3205 . . . . . . . 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 457 . . . . . . 7  |-  ( ( y  e.  u  /\  A. x  e.  u  ( ~P x  C_  u  /\  ~P x  e.  u
) )  ->  ~P y  C_  u )
8 rabss 3563 . . . . . . . 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 3109 . . . . . . . . . . 11  |-  y  e. 
_V
1110canth2 7663 . . . . . . . . . 10  |-  y  ~<  ~P y
12 sdomdom 7536 . . . . . . . . . 10  |-  ( y 
~<  ~P y  ->  y  ~<_  ~P y )
1311, 12ax-mp 5 . . . . . . . . 9  |-  y  ~<_  ~P y
14 vex 3109 . . . . . . . . . 10  |-  u  e. 
_V
15 ssdomg 7554 . . . . . . . . . 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 7561 . . . . . . . . 9  |-  ( ( y  ~<_  ~P y  /\  ~P y  ~<_  u )  -> 
y  ~<_  u )
1813, 16, 17sylancr 661 . . . . . . . 8  |-  ( ~P y  C_  u  ->  y  ~<_  u )
19 tskwe 8322 . . . . . . . . 9  |-  ( ( u  e.  _V  /\  { x  e.  ~P u  |  x  ~<  u }  C_  u )  ->  u  e.  dom  card )
2014, 19mpan 668 . . . . . . . 8  |-  ( { x  e.  ~P u  |  x  ~<  u }  C_  u  ->  u  e.  dom  card )
21 numdom 8410 . . . . . . . . 9  |-  ( ( u  e.  dom  card  /\  y  ~<_  u )  -> 
y  e.  dom  card )
2221expcom 433 . . . . . . . 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 1191 . . . . 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 1727 . . . 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 2450 1  |-  dom  card  =  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   class class class wbr 4439   dom cdm 4988    ~<_ cdom 7507    ~< csdm 7508   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-groth 9190
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-recs 7034  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-card 8311
This theorem is referenced by:  axgroth3  9198
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