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Theorem ttac 30953
Description: Tarski's theorem about choice: infxpidm 8940 is equivalent to ax-ac 8842. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
Assertion
Ref Expression
ttac  |-  (CHOICE  <->  A. c
( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)

Proof of Theorem ttac
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dfac10 8520 . 2  |-  (CHOICE  <->  dom  card  =  _V )
2 vex 3098 . . . . . 6  |-  c  e. 
_V
3 eleq2 2516 . . . . . 6  |-  ( dom 
card  =  _V  ->  ( c  e.  dom  card  <->  c  e.  _V ) )
42, 3mpbiri 233 . . . . 5  |-  ( dom 
card  =  _V  ->  c  e.  dom  card )
5 infxpidm2 8397 . . . . . 6  |-  ( ( c  e.  dom  card  /\ 
om  ~<_  c )  -> 
( c  X.  c
)  ~~  c )
65ex 434 . . . . 5  |-  ( c  e.  dom  card  ->  ( om  ~<_  c  ->  (
c  X.  c ) 
~~  c ) )
74, 6syl 16 . . . 4  |-  ( dom 
card  =  _V  ->  ( om  ~<_  c  ->  (
c  X.  c ) 
~~  c ) )
87alrimiv 1706 . . 3  |-  ( dom 
card  =  _V  ->  A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
9 finnum 8332 . . . . . . 7  |-  ( a  e.  Fin  ->  a  e.  dom  card )
109adantl 466 . . . . . 6  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  a  e.  Fin )  ->  a  e. 
dom  card )
11 harcl 7990 . . . . . . . . 9  |-  (har `  a )  e.  On
12 onenon 8333 . . . . . . . . 9  |-  ( (har
`  a )  e.  On  ->  (har `  a
)  e.  dom  card )
1311, 12ax-mp 5 . . . . . . . 8  |-  (har `  a )  e.  dom  card
14 fvex 5866 . . . . . . . . . . . . . 14  |-  (har `  a )  e.  _V
15 vex 3098 . . . . . . . . . . . . . 14  |-  a  e. 
_V
1614, 15unex 6583 . . . . . . . . . . . . 13  |-  ( (har
`  a )  u.  a )  e.  _V
17 harinf 30951 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  _V  /\  -.  a  e.  Fin )  ->  om  C_  (har `  a ) )
1815, 17mpan 670 . . . . . . . . . . . . . 14  |-  ( -.  a  e.  Fin  ->  om  C_  (har `  a )
)
19 ssun1 3652 . . . . . . . . . . . . . 14  |-  (har `  a )  C_  (
(har `  a )  u.  a )
2018, 19syl6ss 3501 . . . . . . . . . . . . 13  |-  ( -.  a  e.  Fin  ->  om  C_  ( (har `  a
)  u.  a ) )
21 ssdomg 7563 . . . . . . . . . . . . 13  |-  ( ( (har `  a )  u.  a )  e.  _V  ->  ( om  C_  (
(har `  a )  u.  a )  ->  om  ~<_  ( (har
`  a )  u.  a ) ) )
2216, 20, 21mpsyl 63 . . . . . . . . . . . 12  |-  ( -.  a  e.  Fin  ->  om  ~<_  ( (har `  a
)  u.  a ) )
23 breq2 4441 . . . . . . . . . . . . . 14  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( om  ~<_  c 
<->  om  ~<_  ( (har `  a )  u.  a
) ) )
24 xpeq12 5008 . . . . . . . . . . . . . . . 16  |-  ( ( c  =  ( (har
`  a )  u.  a )  /\  c  =  ( (har `  a )  u.  a
) )  ->  (
c  X.  c )  =  ( ( (har
`  a )  u.  a )  X.  (
(har `  a )  u.  a ) ) )
2524anidms 645 . . . . . . . . . . . . . . 15  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( c  X.  c )  =  ( ( (har `  a
)  u.  a )  X.  ( (har `  a )  u.  a
) ) )
26 id 22 . . . . . . . . . . . . . . 15  |-  ( c  =  ( (har `  a )  u.  a
)  ->  c  =  ( (har `  a )  u.  a ) )
2725, 26breq12d 4450 . . . . . . . . . . . . . 14  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( (
c  X.  c ) 
~~  c  <->  ( (
(har `  a )  u.  a )  X.  (
(har `  a )  u.  a ) )  ~~  ( (har `  a )  u.  a ) ) )
2823, 27imbi12d 320 . . . . . . . . . . . . 13  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( ( om 
~<_  c  ->  ( c  X.  c )  ~~  c )  <->  ( om  ~<_  ( (har `  a )  u.  a )  ->  (
( (har `  a
)  u.  a )  X.  ( (har `  a )  u.  a
) )  ~~  (
(har `  a )  u.  a ) ) ) )
2916, 28spcv 3186 . . . . . . . . . . . 12  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  ( om  ~<_  ( (har
`  a )  u.  a )  ->  (
( (har `  a
)  u.  a )  X.  ( (har `  a )  u.  a
) )  ~~  (
(har `  a )  u.  a ) ) )
3022, 29syl5 32 . . . . . . . . . . 11  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  ( -.  a  e. 
Fin  ->  ( ( (har
`  a )  u.  a )  X.  (
(har `  a )  u.  a ) )  ~~  ( (har `  a )  u.  a ) ) )
3130imp 429 . . . . . . . . . 10  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  (
( (har `  a
)  u.  a )  X.  ( (har `  a )  u.  a
) )  ~~  (
(har `  a )  u.  a ) )
32 harndom 7993 . . . . . . . . . . . 12  |-  -.  (har `  a )  ~<_  a
33 ssdomg 7563 . . . . . . . . . . . . . 14  |-  ( ( (har `  a )  u.  a )  e.  _V  ->  ( (har `  a
)  C_  ( (har `  a )  u.  a
)  ->  (har `  a
)  ~<_  ( (har `  a )  u.  a
) ) )
3416, 19, 33mp2 9 . . . . . . . . . . . . 13  |-  (har `  a )  ~<_  ( (har
`  a )  u.  a )
35 domtr 7570 . . . . . . . . . . . . 13  |-  ( ( (har `  a )  ~<_  ( (har `  a )  u.  a )  /\  (
(har `  a )  u.  a )  ~<_  a )  ->  (har `  a
)  ~<_  a )
3634, 35mpan 670 . . . . . . . . . . . 12  |-  ( ( (har `  a )  u.  a )  ~<_  a  -> 
(har `  a )  ~<_  a )
3732, 36mto 176 . . . . . . . . . . 11  |-  -.  (
(har `  a )  u.  a )  ~<_  a
38 unxpwdom2 8017 . . . . . . . . . . 11  |-  ( ( ( (har `  a
)  u.  a )  X.  ( (har `  a )  u.  a
) )  ~~  (
(har `  a )  u.  a )  ->  (
( (har `  a
)  u.  a )  ~<_*  (har `  a )  \/  ( (har `  a
)  u.  a )  ~<_  a ) )
39 orel2 383 . . . . . . . . . . 11  |-  ( -.  ( (har `  a
)  u.  a )  ~<_  a  ->  ( (
( (har `  a
)  u.  a )  ~<_*  (har `  a )  \/  ( (har `  a
)  u.  a )  ~<_  a )  ->  (
(har `  a )  u.  a )  ~<_*  (har `  a )
) )
4037, 38, 39mpsyl 63 . . . . . . . . . 10  |-  ( ( ( (har `  a
)  u.  a )  X.  ( (har `  a )  u.  a
) )  ~~  (
(har `  a )  u.  a )  ->  (
(har `  a )  u.  a )  ~<_*  (har `  a )
)
4131, 40syl 16 . . . . . . . . 9  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  (
(har `  a )  u.  a )  ~<_*  (har `  a )
)
42 wdomnumr 8448 . . . . . . . . . 10  |-  ( (har
`  a )  e. 
dom  card  ->  ( (
(har `  a )  u.  a )  ~<_*  (har `  a )  <->  ( (har `  a )  u.  a )  ~<_  (har `  a ) ) )
4313, 42ax-mp 5 . . . . . . . . 9  |-  ( ( (har `  a )  u.  a )  ~<_*  (har `  a )  <->  ( (har `  a )  u.  a )  ~<_  (har `  a ) )
4441, 43sylib 196 . . . . . . . 8  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  (
(har `  a )  u.  a )  ~<_  (har `  a ) )
45 numdom 8422 . . . . . . . 8  |-  ( ( (har `  a )  e.  dom  card  /\  (
(har `  a )  u.  a )  ~<_  (har `  a ) )  -> 
( (har `  a
)  u.  a )  e.  dom  card )
4613, 44, 45sylancr 663 . . . . . . 7  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  (
(har `  a )  u.  a )  e.  dom  card )
47 ssun2 3653 . . . . . . 7  |-  a  C_  ( (har `  a )  u.  a )
48 ssnum 8423 . . . . . . 7  |-  ( ( ( (har `  a
)  u.  a )  e.  dom  card  /\  a  C_  ( (har `  a
)  u.  a ) )  ->  a  e.  dom  card )
4946, 47, 48sylancl 662 . . . . . 6  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  a  e.  dom  card )
5010, 49pm2.61dan 791 . . . . 5  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  a  e.  dom  card )
5150alrimiv 1706 . . . 4  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  A. a  a  e. 
dom  card )
52 eqv 3787 . . . 4  |-  ( dom 
card  =  _V  <->  A. a 
a  e.  dom  card )
5351, 52sylibr 212 . . 3  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  dom  card  =  _V )
548, 53impbii 188 . 2  |-  ( dom 
card  =  _V  <->  A. c
( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
551, 54bitri 249 1  |-  (CHOICE  <->  A. c
( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1381    = wceq 1383    e. wcel 1804   _Vcvv 3095    u. cun 3459    C_ wss 3461   class class class wbr 4437   Oncon0 4868    X. cxp 4987   dom cdm 4989   ` cfv 5578   omcom 6685    ~~ cen 7515    ~<_ cdom 7516   Fincfn 7518  harchar 7985    ~<_* cwdom 7986   cardccrd 8319  CHOICEwac 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-har 7987  df-wdom 7988  df-card 8323  df-acn 8326  df-ac 8500
This theorem is referenced by: (None)
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