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Theorem ttac 29356
Description: Tarski's theorem about choice: infxpidm 8718 is equivalent to ax-ac 8620. (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 8298 . 2  |-  (CHOICE  <->  dom  card  =  _V )
2 vex 2970 . . . . . 6  |-  c  e. 
_V
3 eleq2 2499 . . . . . 6  |-  ( dom 
card  =  _V  ->  ( c  e.  dom  card  <->  c  e.  _V ) )
42, 3mpbiri 233 . . . . 5  |-  ( dom 
card  =  _V  ->  c  e.  dom  card )
5 infxpidm2 8175 . . . . . 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 1685 . . 3  |-  ( dom 
card  =  _V  ->  A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )
)
9 finnum 8110 . . . . . . 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 7768 . . . . . . . . 9  |-  (har `  a )  e.  On
12 onenon 8111 . . . . . . . . 9  |-  ( (har
`  a )  e.  On  ->  (har `  a
)  e.  dom  card )
1311, 12ax-mp 5 . . . . . . . 8  |-  (har `  a )  e.  dom  card
14 fvex 5696 . . . . . . . . . . . . . 14  |-  (har `  a )  e.  _V
15 vex 2970 . . . . . . . . . . . . . 14  |-  a  e. 
_V
1614, 15unex 6373 . . . . . . . . . . . . 13  |-  ( (har
`  a )  u.  a )  e.  _V
17 harinf 29354 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  _V  /\  -.  a  e.  Fin )  ->  om  C_  (har `  a ) )
1815, 17mpan 670 . . . . . . . . . . . . . 14  |-  ( -.  a  e.  Fin  ->  om  C_  (har `  a )
)
19 ssun1 3514 . . . . . . . . . . . . . 14  |-  (har `  a )  C_  (
(har `  a )  u.  a )
2018, 19syl6ss 3363 . . . . . . . . . . . . 13  |-  ( -.  a  e.  Fin  ->  om  C_  ( (har `  a
)  u.  a ) )
21 ssdomg 7347 . . . . . . . . . . . . 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 4291 . . . . . . . . . . . . . 14  |-  ( c  =  ( (har `  a )  u.  a
)  ->  ( om  ~<_  c 
<->  om  ~<_  ( (har `  a )  u.  a
) ) )
24 xpeq12 4854 . . . . . . . . . . . . . . . 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 4300 . . . . . . . . . . . . . 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 3058 . . . . . . . . . . . 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 7771 . . . . . . . . . . . 12  |-  -.  (har `  a )  ~<_  a
33 ssdomg 7347 . . . . . . . . . . . . . 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 7354 . . . . . . . . . . . . 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 7795 . . . . . . . . . . 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 8226 . . . . . . . . . 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 8200 . . . . . . . 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 3515 . . . . . . 7  |-  a  C_  ( (har `  a )  u.  a )
48 ssnum 8201 . . . . . . 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 789 . . . . 5  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  a  e.  dom  card )
5150alrimiv 1685 . . . 4  |-  ( A. c ( om  ~<_  c  -> 
( c  X.  c
)  ~~  c )  ->  A. a  a  e. 
dom  card )
52 eqv 3648 . . . 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 1367    = wceq 1369    e. wcel 1756   _Vcvv 2967    u. cun 3321    C_ wss 3323   class class class wbr 4287   Oncon0 4714    X. cxp 4833   dom cdm 4835   ` cfv 5413   omcom 6471    ~~ cen 7299    ~<_ cdom 7300   Fincfn 7302  harchar 7763    ~<_* cwdom 7764   cardccrd 8097  CHOICEwac 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-oi 7716  df-har 7765  df-wdom 7766  df-card 8101  df-acn 8104  df-ac 8278
This theorem is referenced by: (None)
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