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Theorem ttac 26997
Description: Tarski's theorem about choice: infxpidm 8393 is equivalent to ax-ac 8295. (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 7973 . 2  |-  (CHOICE  <->  dom  card  =  _V )
2 vex 2919 . . . . . 6  |-  c  e. 
_V
3 eleq2 2465 . . . . . 6  |-  ( dom 
card  =  _V  ->  ( c  e.  dom  card  <->  c  e.  _V ) )
42, 3mpbiri 225 . . . . 5  |-  ( dom 
card  =  _V  ->  c  e.  dom  card )
5 infxpidm2 7854 . . . . . 6  |-  ( ( c  e.  dom  card  /\ 
om  ~<_  c )  -> 
( c  X.  c
)  ~~  c )
65ex 424 . . . . 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 1638 . . 3  |-  ( dom 
card  =  _V  ->  A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
9 finnum 7791 . . . . . . 7  |-  ( a  e.  Fin  ->  a  e.  dom  card )
109adantl 453 . . . . . 6  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  a  e.  Fin )  ->  a  e. 
dom  card )
11 harcl 7485 . . . . . . . . 9  |-  (har `  a )  e.  On
12 onenon 7792 . . . . . . . . 9  |-  ( (har
`  a )  e.  On  ->  (har `  a
)  e.  dom  card )
1311, 12ax-mp 8 . . . . . . . 8  |-  (har `  a )  e.  dom  card
14 fvex 5701 . . . . . . . . . . . . . 14  |-  (har `  a )  e.  _V
15 vex 2919 . . . . . . . . . . . . . 14  |-  a  e. 
_V
1614, 15unex 4666 . . . . . . . . . . . . 13  |-  ( (har
`  a )  u.  a )  e.  _V
17 harinf 26995 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  _V  /\  -.  a  e.  Fin )  ->  om  C_  (har `  a ) )
1815, 17mpan 652 . . . . . . . . . . . . . 14  |-  ( -.  a  e.  Fin  ->  om  C_  (har `  a )
)
19 ssun1 3470 . . . . . . . . . . . . . 14  |-  (har `  a )  C_  (
(har `  a )  u.  a )
2018, 19syl6ss 3320 . . . . . . . . . . . . 13  |-  ( -.  a  e.  Fin  ->  om  C_  ( (har `  a
)  u.  a ) )
21 ssdomg 7112 . . . . . . . . . . . . 13  |-  ( ( (har `  a )  u.  a )  e.  _V  ->  ( om  C_  (
(har `  a )  u.  a )  ->  om  ~<_  ( (har
`  a )  u.  a ) ) )
2216, 20, 21mpsyl 61 . . . . . . . . . . . 12  |-  ( -.  a  e.  Fin  ->  om  ~<_  ( (har `  a
)  u.  a ) )
23 breq2 4176 . . . . . . . . . . . . . 14  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( om  ~<_  c 
<->  om  ~<_  ( (har `  a )  u.  a
) ) )
24 xpeq12 4856 . . . . . . . . . . . . . . . 16  |-  ( ( c  =  ( (har
`  a )  u.  a )  /\  c  =  ( (har `  a )  u.  a
) )  ->  (
c  X.  c )  =  ( ( (har
`  a )  u.  a )  X.  (
(har `  a )  u.  a ) ) )
2524anidms 627 . . . . . . . . . . . . . . 15  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( c  X.  c )  =  ( ( (har `  a
)  u.  a )  X.  ( (har `  a )  u.  a
) ) )
26 id 20 . . . . . . . . . . . . . . 15  |-  ( c  =  ( (har `  a )  u.  a
)  ->  c  =  ( (har `  a )  u.  a ) )
2725, 26breq12d 4185 . . . . . . . . . . . . . 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 312 . . . . . . . . . . . . 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 3002 . . . . . . . . . . . 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 30 . . . . . . . . . . 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 419 . . . . . . . . . 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 7488 . . . . . . . . . . . 12  |-  -.  (har `  a )  ~<_  a
33 ssdomg 7112 . . . . . . . . . . . . . 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 7119 . . . . . . . . . . . . 13  |-  ( ( (har `  a )  ~<_  ( (har `  a )  u.  a )  /\  (
(har `  a )  u.  a )  ~<_  a )  ->  (har `  a
)  ~<_  a )
3634, 35mpan 652 . . . . . . . . . . . 12  |-  ( ( (har `  a )  u.  a )  ~<_  a  -> 
(har `  a )  ~<_  a )
3732, 36mto 169 . . . . . . . . . . 11  |-  -.  (
(har `  a )  u.  a )  ~<_  a
38 unxpwdom2 7512 . . . . . . . . . . 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 373 . . . . . . . . . . 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 61 . . . . . . . . . 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 7901 . . . . . . . . . 10  |-  ( (har
`  a )  e. 
dom  card  ->  ( (
(har `  a )  u.  a )  ~<_*  (har `  a )  <->  ( (har `  a )  u.  a )  ~<_  (har `  a ) ) )
4313, 42ax-mp 8 . . . . . . . . 9  |-  ( ( (har `  a )  u.  a )  ~<_*  (har `  a )  <->  ( (har `  a )  u.  a )  ~<_  (har `  a ) )
4441, 43sylib 189 . . . . . . . 8  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  (
(har `  a )  u.  a )  ~<_  (har `  a ) )
45 numdom 7875 . . . . . . . 8  |-  ( ( (har `  a )  e.  dom  card  /\  (
(har `  a )  u.  a )  ~<_  (har `  a ) )  -> 
( (har `  a
)  u.  a )  e.  dom  card )
4613, 44, 45sylancr 645 . . . . . . 7  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  (
(har `  a )  u.  a )  e.  dom  card )
47 ssun2 3471 . . . . . . 7  |-  a  C_  ( (har `  a )  u.  a )
48 ssnum 7876 . . . . . . 7  |-  ( ( ( (har `  a
)  u.  a )  e.  dom  card  /\  a  C_  ( (har `  a
)  u.  a ) )  ->  a  e.  dom  card )
4946, 47, 48sylancl 644 . . . . . 6  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  a  e.  dom  card )
5010, 49pm2.61dan 767 . . . . 5  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  a  e.  dom  card )
5150alrimiv 1638 . . . 4  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  A. a  a  e. 
dom  card )
52 eqv 3603 . . . 4  |-  ( dom 
card  =  _V  <->  A. a 
a  e.  dom  card )
5351, 52sylibr 204 . . 3  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  dom  card  =  _V )
548, 53impbii 181 . 2  |-  ( dom 
card  =  _V  <->  A. c
( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
551, 54bitri 241 1  |-  (CHOICE  <->  A. c
( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    C_ wss 3280   class class class wbr 4172   Oncon0 4541   omcom 4804    X. cxp 4835   dom cdm 4837   ` cfv 5413    ~~ cen 7065    ~<_ cdom 7066   Fincfn 7068  harchar 7480    ~<_* cwdom 7481   cardccrd 7778  CHOICEwac 7952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-wdom 7483  df-card 7782  df-acn 7785  df-ac 7953
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