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Theorem ttac 31144
Description: Tarski's theorem about choice: infxpidm 8850 is equivalent to ax-ac 8752. (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 8430 . 2  |-  (CHOICE  <->  dom  card  =  _V )
2 vex 3037 . . . . . 6  |-  c  e. 
_V
3 eleq2 2455 . . . . . 6  |-  ( dom 
card  =  _V  ->  ( c  e.  dom  card  <->  c  e.  _V ) )
42, 3mpbiri 233 . . . . 5  |-  ( dom 
card  =  _V  ->  c  e.  dom  card )
5 infxpidm2 8307 . . . . . 6  |-  ( ( c  e.  dom  card  /\ 
om  ~<_  c )  -> 
( c  X.  c
)  ~~  c )
65ex 432 . . . . 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 1727 . . 3  |-  ( dom 
card  =  _V  ->  A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
9 finnum 8242 . . . . . . 7  |-  ( a  e.  Fin  ->  a  e.  dom  card )
109adantl 464 . . . . . 6  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  a  e.  Fin )  ->  a  e. 
dom  card )
11 harcl 7902 . . . . . . . . 9  |-  (har `  a )  e.  On
12 onenon 8243 . . . . . . . . 9  |-  ( (har
`  a )  e.  On  ->  (har `  a
)  e.  dom  card )
1311, 12ax-mp 5 . . . . . . . 8  |-  (har `  a )  e.  dom  card
14 fvex 5784 . . . . . . . . . . . . . 14  |-  (har `  a )  e.  _V
15 vex 3037 . . . . . . . . . . . . . 14  |-  a  e. 
_V
1614, 15unex 6497 . . . . . . . . . . . . 13  |-  ( (har
`  a )  u.  a )  e.  _V
17 harinf 31142 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  _V  /\  -.  a  e.  Fin )  ->  om  C_  (har `  a ) )
1815, 17mpan 668 . . . . . . . . . . . . . 14  |-  ( -.  a  e.  Fin  ->  om  C_  (har `  a )
)
19 ssun1 3581 . . . . . . . . . . . . . 14  |-  (har `  a )  C_  (
(har `  a )  u.  a )
2018, 19syl6ss 3429 . . . . . . . . . . . . 13  |-  ( -.  a  e.  Fin  ->  om  C_  ( (har `  a
)  u.  a ) )
21 ssdomg 7480 . . . . . . . . . . . . 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 4371 . . . . . . . . . . . . . 14  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( om  ~<_  c 
<->  om  ~<_  ( (har `  a )  u.  a
) ) )
24 xpeq12 4932 . . . . . . . . . . . . . . . 16  |-  ( ( c  =  ( (har
`  a )  u.  a )  /\  c  =  ( (har `  a )  u.  a
) )  ->  (
c  X.  c )  =  ( ( (har
`  a )  u.  a )  X.  (
(har `  a )  u.  a ) ) )
2524anidms 643 . . . . . . . . . . . . . . 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 4380 . . . . . . . . . . . . . 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 318 . . . . . . . . . . . . 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 3125 . . . . . . . . . . . 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 427 . . . . . . . . . 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 7905 . . . . . . . . . . . 12  |-  -.  (har `  a )  ~<_  a
33 ssdomg 7480 . . . . . . . . . . . . . 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 7487 . . . . . . . . . . . . 13  |-  ( ( (har `  a )  ~<_  ( (har `  a )  u.  a )  /\  (
(har `  a )  u.  a )  ~<_  a )  ->  (har `  a
)  ~<_  a )
3634, 35mpan 668 . . . . . . . . . . . 12  |-  ( ( (har `  a )  u.  a )  ~<_  a  -> 
(har `  a )  ~<_  a )
3732, 36mto 176 . . . . . . . . . . 11  |-  -.  (
(har `  a )  u.  a )  ~<_  a
38 unxpwdom2 7929 . . . . . . . . . . 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 381 . . . . . . . . . . 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 8358 . . . . . . . . . 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 8332 . . . . . . . 8  |-  ( ( (har `  a )  e.  dom  card  /\  (
(har `  a )  u.  a )  ~<_  (har `  a ) )  -> 
( (har `  a
)  u.  a )  e.  dom  card )
4613, 44, 45sylancr 661 . . . . . . 7  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  (
(har `  a )  u.  a )  e.  dom  card )
47 ssun2 3582 . . . . . . 7  |-  a  C_  ( (har `  a )  u.  a )
48 ssnum 8333 . . . . . . 7  |-  ( ( ( (har `  a
)  u.  a )  e.  dom  card  /\  a  C_  ( (har `  a
)  u.  a ) )  ->  a  e.  dom  card )
4946, 47, 48sylancl 660 . . . . . 6  |-  ( ( A. c ( om  ~<_  c  ->  ( c  X.  c )  ~~  c
)  /\  -.  a  e.  Fin )  ->  a  e.  dom  card )
5010, 49pm2.61dan 789 . . . . 5  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  a  e.  dom  card )
5150alrimiv 1727 . . . 4  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  A. a  a  e. 
dom  card )
52 eqv 3728 . . . 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 366    /\ wa 367   A.wal 1397    = wceq 1399    e. wcel 1826   _Vcvv 3034    u. cun 3387    C_ wss 3389   class class class wbr 4367   Oncon0 4792    X. cxp 4911   dom cdm 4913   ` cfv 5496   omcom 6599    ~~ cen 7432    ~<_ cdom 7433   Fincfn 7435  harchar 7897    ~<_* cwdom 7898   cardccrd 8229  CHOICEwac 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-oi 7850  df-har 7899  df-wdom 7900  df-card 8233  df-acn 8236  df-ac 8410
This theorem is referenced by: (None)
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