MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac13 Structured version   Visualization version   Unicode version

Theorem dfac13 8577
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 3050 . . . 4  |-  x  e. 
_V
2 acacni 8575 . . . . 5  |-  ( (CHOICE  /\  x  e.  _V )  -> AC  x  =  _V )
31, 2mpan2 678 . . . 4  |-  (CHOICE  -> AC  x  =  _V )
41, 3syl5eleqr 2538 . . 3  |-  (CHOICE  ->  x  e. AC  x )
54alrimiv 1775 . 2  |-  (CHOICE  ->  A. x  x  e. AC  x )
6 vex 3050 . . . . . . . . 9  |-  z  e. 
_V
76pwex 4589 . . . . . . . 8  |-  ~P z  e.  _V
8 id 22 . . . . . . . . 9  |-  ( x  =  ~P z  ->  x  =  ~P z
)
9 acneq 8479 . . . . . . . . 9  |-  ( x  =  ~P z  -> AC  x  = AC  ~P z )
108, 9eleq12d 2525 . . . . . . . 8  |-  ( x  =  ~P z  -> 
( x  e. AC  x  <->  ~P z  e. AC 
~P z ) )
117, 10spcv 3142 . . . . . . 7  |-  ( A. x  x  e. AC  x  ->  ~P z  e. AC  ~P z
)
12 vex 3050 . . . . . . . 8  |-  y  e. 
_V
136canth2 7730 . . . . . . . . . 10  |-  z  ~<  ~P z
14 sdomdom 7602 . . . . . . . . . 10  |-  ( z 
~<  ~P z  ->  z  ~<_  ~P z )
15 acndom2 8490 . . . . . . . . . 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 8488 . . . . . . . . 9  |-  ( z  e. AC  ~P z  <->  z  e.  dom  card )
1816, 17sylib 200 . . . . . . . 8  |-  ( ~P z  e. AC  ~P z  ->  z  e.  dom  card )
19 numacn 8485 . . . . . . . 8  |-  ( y  e.  _V  ->  (
z  e.  dom  card  -> 
z  e. AC  y ) )
2012, 18, 19mpsyl 65 . . . . . . 7  |-  ( ~P z  e. AC  ~P z  ->  z  e. AC  y )
2111, 20syl 17 . . . . . 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 244 . . . . 5  |-  ( A. x  x  e. AC  x  -> 
( z  e. AC  y  <->  z  e.  _V ) )
2423eqrdv 2451 . . . 4  |-  ( A. x  x  e. AC  x  -> AC  y  =  _V )
2524alrimiv 1775 . . 3  |-  ( A. x  x  e. AC  x  ->  A. yAC  y  =  _V )
26 dfacacn 8576 . . 3  |-  (CHOICE  <->  A. yAC  y  =  _V )
2725, 26sylibr 216 . 2  |-  ( A. x  x  e. AC  x  -> CHOICE )
285, 27impbii 191 1  |-  (CHOICE  <->  A. x  x  e. AC  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1444    = wceq 1446    e. wcel 1889   _Vcvv 3047   ~Pcpw 3953   class class class wbr 4405   dom cdm 4837    ~<_ cdom 7572    ~< csdm 7573   cardccrd 8374  AC wacn 8377  CHOICEwac 8551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-1o 7187  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-acn 8381  df-ac 8552
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator