MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grothac Structured version   Unicode version

Theorem grothac 9103
Description: The Tarski-Grothendieck Axiom implies the Axiom of Choice (in the form of cardeqv 8744). This can be put in a more conventional form via ween 8311 and dfac8 8410. 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 9101 . . . 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 3966 . . . . . . . . . . 11  |-  ( x  =  y  ->  ~P x  =  ~P y
)
32sseq1d 3486 . . . . . . . . . 10  |-  ( x  =  y  ->  ( ~P x  C_  u  <->  ~P y  C_  u ) )
42eleq1d 2521 . . . . . . . . . 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 3171 . . . . . . . 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 3532 . . . . . . . 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 3075 . . . . . . . . . . 11  |-  y  e. 
_V
1110canth2 7569 . . . . . . . . . 10  |-  y  ~<  ~P y
12 sdomdom 7442 . . . . . . . . . 10  |-  ( y 
~<  ~P y  ->  y  ~<_  ~P y )
1311, 12ax-mp 5 . . . . . . . . 9  |-  y  ~<_  ~P y
14 vex 3075 . . . . . . . . . 10  |-  u  e. 
_V
15 ssdomg 7460 . . . . . . . . . 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 7467 . . . . . . . . 9  |-  ( ( y  ~<_  ~P y  /\  ~P y  ~<_  u )  -> 
y  ~<_  u )
1813, 16, 17sylancr 663 . . . . . . . 8  |-  ( ~P y  C_  u  ->  y  ~<_  u )
19 tskwe 8226 . . . . . . . . 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 8314 . . . . . . . . 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 1185 . . . . 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 1689 . . . 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 2448 1  |-  dom  card  =  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   A.wral 2796   {crab 2800   _Vcvv 3072    C_ wss 3431   ~Pcpw 3963   class class class wbr 4395   dom cdm 4943    ~<_ cdom 7413    ~< csdm 7414   cardccrd 8211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-groth 9096
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-recs 6937  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-card 8215
This theorem is referenced by:  axgroth3  9104
  Copyright terms: Public domain W3C validator