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Theorem dfac13 8539
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 3112 . . . 4  |-  x  e. 
_V
2 acacni 8537 . . . . 5  |-  ( (CHOICE  /\  x  e.  _V )  -> AC  x  =  _V )
31, 2mpan2 671 . . . 4  |-  (CHOICE  -> AC  x  =  _V )
41, 3syl5eleqr 2552 . . 3  |-  (CHOICE  ->  x  e. AC  x )
54alrimiv 1720 . 2  |-  (CHOICE  ->  A. x  x  e. AC  x )
6 vex 3112 . . . . . . . . 9  |-  z  e. 
_V
76pwex 4639 . . . . . . . 8  |-  ~P z  e.  _V
8 id 22 . . . . . . . . 9  |-  ( x  =  ~P z  ->  x  =  ~P z
)
9 acneq 8441 . . . . . . . . 9  |-  ( x  =  ~P z  -> AC  x  = AC  ~P z )
108, 9eleq12d 2539 . . . . . . . 8  |-  ( x  =  ~P z  -> 
( x  e. AC  x  <->  ~P z  e. AC 
~P z ) )
117, 10spcv 3200 . . . . . . 7  |-  ( A. x  x  e. AC  x  ->  ~P z  e. AC  ~P z
)
12 vex 3112 . . . . . . . 8  |-  y  e. 
_V
136canth2 7689 . . . . . . . . . 10  |-  z  ~<  ~P z
14 sdomdom 7562 . . . . . . . . . 10  |-  ( z 
~<  ~P z  ->  z  ~<_  ~P z )
15 acndom2 8452 . . . . . . . . . 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 8450 . . . . . . . . 9  |-  ( z  e. AC  ~P z  <->  z  e.  dom  card )
1816, 17sylib 196 . . . . . . . 8  |-  ( ~P z  e. AC  ~P z  ->  z  e.  dom  card )
19 numacn 8447 . . . . . . . 8  |-  ( y  e.  _V  ->  (
z  e.  dom  card  -> 
z  e. AC  y ) )
2012, 18, 19mpsyl 63 . . . . . . 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 240 . . . . 5  |-  ( A. x  x  e. AC  x  -> 
( z  e. AC  y  <->  z  e.  _V ) )
2423eqrdv 2454 . . . 4  |-  ( A. x  x  e. AC  x  -> AC  y  =  _V )
2524alrimiv 1720 . . 3  |-  ( A. x  x  e. AC  x  ->  A. yAC  y  =  _V )
26 dfacacn 8538 . . 3  |-  (CHOICE  <->  A. yAC  y  =  _V )
2725, 26sylibr 212 . 2  |-  ( A. x  x  e. AC  x  -> CHOICE )
285, 27impbii 188 1  |-  (CHOICE  <->  A. x  x  e. AC  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1393    = wceq 1395    e. wcel 1819   _Vcvv 3109   ~Pcpw 4015   class class class wbr 4456   dom cdm 5008    ~<_ cdom 7533    ~< csdm 7534   cardccrd 8333  AC wacn 8336  CHOICEwac 8513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-acn 8340  df-ac 8514
This theorem is referenced by: (None)
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