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Theorem dfac13 8024
Description: The axiom of choice holds iff every set has choice sequences as long as itself. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
dfac13  |-  (CHOICE  <->  A. x  x  e. AC  x )

Proof of Theorem dfac13
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2961 . . . 4  |-  x  e. 
_V
2 acacni 8022 . . . . 5  |-  ( (CHOICE  /\  x  e.  _V )  -> AC  x  =  _V )
31, 2mpan2 654 . . . 4  |-  (CHOICE  -> AC  x  =  _V )
41, 3syl5eleqr 2525 . . 3  |-  (CHOICE  ->  x  e. AC  x )
54alrimiv 1642 . 2  |-  (CHOICE  ->  A. x  x  e. AC  x )
6 vex 2961 . . . . . . . . 9  |-  z  e. 
_V
76pwex 4384 . . . . . . . 8  |-  ~P z  e.  _V
8 id 21 . . . . . . . . 9  |-  ( x  =  ~P z  ->  x  =  ~P z
)
9 acneq 7926 . . . . . . . . 9  |-  ( x  =  ~P z  -> AC  x  = AC  ~P z )
108, 9eleq12d 2506 . . . . . . . 8  |-  ( x  =  ~P z  -> 
( x  e. AC  x  <->  ~P z  e. AC 
~P z ) )
117, 10spcv 3044 . . . . . . 7  |-  ( A. x  x  e. AC  x  ->  ~P z  e. AC  ~P z
)
12 vex 2961 . . . . . . . 8  |-  y  e. 
_V
136canth2 7262 . . . . . . . . . 10  |-  z  ~<  ~P z
14 sdomdom 7137 . . . . . . . . . 10  |-  ( z 
~<  ~P z  ->  z  ~<_  ~P z )
15 acndom2 7937 . . . . . . . . . 10  |-  ( z  ~<_  ~P z  ->  ( ~P z  e. AC  ~P z  ->  z  e. AC  ~P z
) )
1613, 14, 15mp2b 10 . . . . . . . . 9  |-  ( ~P z  e. AC  ~P z  ->  z  e. AC  ~P z
)
17 acnnum 7935 . . . . . . . . 9  |-  ( z  e. AC  ~P z  <->  z  e.  dom  card )
1816, 17sylib 190 . . . . . . . 8  |-  ( ~P z  e. AC  ~P z  ->  z  e.  dom  card )
19 numacn 7932 . . . . . . . 8  |-  ( y  e.  _V  ->  (
z  e.  dom  card  -> 
z  e. AC  y ) )
2012, 18, 19mpsyl 62 . . . . . . 7  |-  ( ~P z  e. AC  ~P z  ->  z  e. AC  y )
2111, 20syl 16 . . . . . 6  |-  ( A. x  x  e. AC  x  -> 
z  e. AC  y )
226a1i 11 . . . . . 6  |-  ( A. x  x  e. AC  x  -> 
z  e.  _V )
2321, 222thd 233 . . . . 5  |-  ( A. x  x  e. AC  x  -> 
( z  e. AC  y  <->  z  e.  _V ) )
2423eqrdv 2436 . . . 4  |-  ( A. x  x  e. AC  x  -> AC  y  =  _V )
2524alrimiv 1642 . . 3  |-  ( A. x  x  e. AC  x  ->  A. yAC  y  =  _V )
26 dfacacn 8023 . . 3  |-  (CHOICE  <->  A. yAC  y  =  _V )
2725, 26sylibr 205 . 2  |-  ( A. x  x  e. AC  x  -> CHOICE )
285, 27impbii 182 1  |-  (CHOICE  <->  A. x  x  e. AC  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2958   ~Pcpw 3801   class class class wbr 4214   dom cdm 4880    ~<_ cdom 7109    ~< csdm 7110   cardccrd 7824  AC wacn 7827  CHOICEwac 7998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-1o 6726  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-acn 7831  df-ac 7999
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