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

Theorem dfac13 8511
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 3109 . . . 4  |-  x  e. 
_V
2 acacni 8509 . . . . 5  |-  ( (CHOICE  /\  x  e.  _V )  -> AC  x  =  _V )
31, 2mpan2 671 . . . 4  |-  (CHOICE  -> AC  x  =  _V )
41, 3syl5eleqr 2555 . . 3  |-  (CHOICE  ->  x  e. AC  x )
54alrimiv 1690 . 2  |-  (CHOICE  ->  A. x  x  e. AC  x )
6 vex 3109 . . . . . . . . 9  |-  z  e. 
_V
76pwex 4623 . . . . . . . 8  |-  ~P z  e.  _V
8 id 22 . . . . . . . . 9  |-  ( x  =  ~P z  ->  x  =  ~P z
)
9 acneq 8413 . . . . . . . . 9  |-  ( x  =  ~P z  -> AC  x  = AC  ~P z )
108, 9eleq12d 2542 . . . . . . . 8  |-  ( x  =  ~P z  -> 
( x  e. AC  x  <->  ~P z  e. AC 
~P z ) )
117, 10spcv 3197 . . . . . . 7  |-  ( A. x  x  e. AC  x  ->  ~P z  e. AC  ~P z
)
12 vex 3109 . . . . . . . 8  |-  y  e. 
_V
136canth2 7660 . . . . . . . . . 10  |-  z  ~<  ~P z
14 sdomdom 7533 . . . . . . . . . 10  |-  ( z 
~<  ~P z  ->  z  ~<_  ~P z )
15 acndom2 8424 . . . . . . . . . 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 8422 . . . . . . . . 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 8419 . . . . . . . 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 2457 . . . 4  |-  ( A. x  x  e. AC  x  -> AC  y  =  _V )
2524alrimiv 1690 . . 3  |-  ( A. x  x  e. AC  x  ->  A. yAC  y  =  _V )
26 dfacacn 8510 . . 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 1372    = wceq 1374    e. wcel 1762   _Vcvv 3106   ~Pcpw 4003   class class class wbr 4440   dom cdm 4992    ~<_ cdom 7504    ~< csdm 7505   cardccrd 8305  AC wacn 8308  CHOICEwac 8485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-1o 7120  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-acn 8312  df-ac 8486
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator