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Theorem ttac 27107
Description: Tarski's theorem about choice: infxpidm 8437 is equivalent to ax-ac 8339. (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 8017 . 2  |-  (CHOICE  <->  dom  card  =  _V )
2 vex 2959 . . . . . 6  |-  c  e. 
_V
3 eleq2 2497 . . . . . 6  |-  ( dom 
card  =  _V  ->  ( c  e.  dom  card  <->  c  e.  _V ) )
42, 3mpbiri 225 . . . . 5  |-  ( dom 
card  =  _V  ->  c  e.  dom  card )
5 infxpidm2 7898 . . . . . 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 1641 . . 3  |-  ( dom 
card  =  _V  ->  A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
9 finnum 7835 . . . . . . 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 7529 . . . . . . . . 9  |-  (har `  a )  e.  On
12 onenon 7836 . . . . . . . . 9  |-  ( (har
`  a )  e.  On  ->  (har `  a
)  e.  dom  card )
1311, 12ax-mp 8 . . . . . . . 8  |-  (har `  a )  e.  dom  card
14 fvex 5742 . . . . . . . . . . . . . 14  |-  (har `  a )  e.  _V
15 vex 2959 . . . . . . . . . . . . . 14  |-  a  e. 
_V
1614, 15unex 4707 . . . . . . . . . . . . 13  |-  ( (har
`  a )  u.  a )  e.  _V
17 harinf 27105 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  _V  /\  -.  a  e.  Fin )  ->  om  C_  (har `  a ) )
1815, 17mpan 652 . . . . . . . . . . . . . 14  |-  ( -.  a  e.  Fin  ->  om  C_  (har `  a )
)
19 ssun1 3510 . . . . . . . . . . . . . 14  |-  (har `  a )  C_  (
(har `  a )  u.  a )
2018, 19syl6ss 3360 . . . . . . . . . . . . 13  |-  ( -.  a  e.  Fin  ->  om  C_  ( (har `  a
)  u.  a ) )
21 ssdomg 7153 . . . . . . . . . . . . 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 4216 . . . . . . . . . . . . . 14  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( om  ~<_  c 
<->  om  ~<_  ( (har `  a )  u.  a
) ) )
24 xpeq12 4897 . . . . . . . . . . . . . . . 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 4225 . . . . . . . . . . . . . 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 3042 . . . . . . . . . . . 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 7532 . . . . . . . . . . . 12  |-  -.  (har `  a )  ~<_  a
33 ssdomg 7153 . . . . . . . . . . . . . 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 7160 . . . . . . . . . . . . 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 7556 . . . . . . . . . . 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 7945 . . . . . . . . . 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 7919 . . . . . . . 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 3511 . . . . . . 7  |-  a  C_  ( (har `  a )  u.  a )
48 ssnum 7920 . . . . . . 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 1641 . . . 4  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  A. a  a  e. 
dom  card )
52 eqv 3643 . . . 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 1549    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318    C_ wss 3320   class class class wbr 4212   Oncon0 4581   omcom 4845    X. cxp 4876   dom cdm 4878   ` cfv 5454    ~~ cen 7106    ~<_ cdom 7107   Fincfn 7109  harchar 7524    ~<_* cwdom 7525   cardccrd 7822  CHOICEwac 7996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-har 7526  df-wdom 7527  df-card 7826  df-acn 7829  df-ac 7997
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