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Theorem ufilen 20994
Description: Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
Assertion
Ref Expression
ufilen  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) A. x  e.  f  x  ~~  X
)
Distinct variable group:    x, f, X

Proof of Theorem ufilen
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 reldom 7601 . . . . . 6  |-  Rel  ~<_
21brrelex2i 4895 . . . . 5  |-  ( om  ~<_  X  ->  X  e.  _V )
3 numth3 8926 . . . . 5  |-  ( X  e.  _V  ->  X  e.  dom  card )
42, 3syl 17 . . . 4  |-  ( om  ~<_  X  ->  X  e.  dom  card )
5 csdfil 20958 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  e.  ( Fil `  X
) )
64, 5mpancom 680 . . 3  |-  ( om  ~<_  X  ->  { y  e.  ~P X  |  ( X  \  y ) 
~<  X }  e.  ( Fil `  X ) )
7 filssufil 20976 . . 3  |-  ( { y  e.  ~P X  |  ( X  \ 
y )  ~<  X }  e.  ( Fil `  X
)  ->  E. f  e.  ( UFil `  X
) { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f
)
86, 7syl 17 . 2  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f
)
9 elfvex 5915 . . . . . . 7  |-  ( f  e.  ( UFil `  X
)  ->  X  e.  _V )
109ad2antlr 738 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  X  e.  _V )
11 ufilfil 20968 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( Fil `  X ) )
12 filelss 20916 . . . . . . . 8  |-  ( ( f  e.  ( Fil `  X )  /\  x  e.  f )  ->  x  C_  X )
1311, 12sylan 478 . . . . . . 7  |-  ( ( f  e.  ( UFil `  X )  /\  x  e.  f )  ->  x  C_  X )
1413adantll 725 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  C_  X )
15 ssdomg 7641 . . . . . 6  |-  ( X  e.  _V  ->  (
x  C_  X  ->  x  ~<_  X ) )
1610, 14, 15sylc 62 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  ~<_  X )
17 filfbas 20912 . . . . . . . . 9  |-  ( f  e.  ( Fil `  X
)  ->  f  e.  ( fBas `  X )
)
1811, 17syl 17 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( fBas `  X )
)
1918adantl 472 . . . . . . 7  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  f  e.  ( fBas `  X
) )
20 fbncp 20903 . . . . . . 7  |-  ( ( f  e.  ( fBas `  X )  /\  x  e.  f )  ->  -.  ( X  \  x
)  e.  f )
2119, 20sylan 478 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  -.  ( X  \  x )  e.  f )
22 difss 3572 . . . . . . . . . . . . . 14  |-  ( X 
\  x )  C_  X
23 elpw2g 4580 . . . . . . . . . . . . . 14  |-  ( X  e.  _V  ->  (
( X  \  x
)  e.  ~P X  <->  ( X  \  x ) 
C_  X ) )
2422, 23mpbiri 241 . . . . . . . . . . . . 13  |-  ( X  e.  _V  ->  ( X  \  x )  e. 
~P X )
25243ad2ant1 1035 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
~P X )
26 simp2 1015 . . . . . . . . . . . . . 14  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  C_  X )
27 dfss4 3689 . . . . . . . . . . . . . 14  |-  ( x 
C_  X  <->  ( X  \  ( X  \  x
) )  =  x )
2826, 27sylib 201 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  =  x )
29 simp3 1016 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  ~<  X )
3028, 29eqbrtrd 4437 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  ~<  X )
31 difeq2 3557 . . . . . . . . . . . . . 14  |-  ( y  =  ( X  \  x )  ->  ( X  \  y )  =  ( X  \  ( X  \  x ) ) )
3231breq1d 4426 . . . . . . . . . . . . 13  |-  ( y  =  ( X  \  x )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( X  \  x
) )  ~<  X ) )
3332elrab 3208 . . . . . . . . . . . 12  |-  ( ( X  \  x )  e.  { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  <->  ( ( X  \  x )  e. 
~P X  /\  ( X  \  ( X  \  x ) )  ~<  X ) )
3425, 30, 33sylanbrc 675 . . . . . . . . . . 11  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
{ y  e.  ~P X  |  ( X  \  y )  ~<  X }
)
35 ssel 3438 . . . . . . . . . . 11  |-  ( { y  e.  ~P X  |  ( X  \ 
y )  ~<  X }  C_  f  ->  ( ( X  \  x )  e. 
{ y  e.  ~P X  |  ( X  \  y )  ~<  X }  ->  ( X  \  x
)  e.  f ) )
3634, 35syl5com 31 . . . . . . . . . 10  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
37363expa 1215 . . . . . . . . 9  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  x  ~<  X )  ->  ( { y  e.  ~P X  | 
( X  \  y
)  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
3837impancom 446 . . . . . . . 8  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  (
x  ~<  X  ->  ( X  \  x )  e.  f ) )
3938con3d 140 . . . . . . 7  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  ( -.  ( X  \  x
)  e.  f  ->  -.  x  ~<  X ) )
4039impancom 446 . . . . . 6  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  -.  ( X  \  x )  e.  f )  ->  ( {
y  e.  ~P X  |  ( X  \ 
y )  ~<  X }  C_  f  ->  -.  x  ~<  X ) )
4110, 14, 21, 40syl21anc 1275 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  -.  x  ~<  X ) )
42 bren2 7626 . . . . . 6  |-  ( x 
~~  X  <->  ( x  ~<_  X  /\  -.  x  ~<  X ) )
4342simplbi2 635 . . . . 5  |-  ( x  ~<_  X  ->  ( -.  x  ~<  X  ->  x  ~~  X ) )
4416, 41, 43sylsyld 58 . . . 4  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  x  ~~  X ) )
4544ralrimdva 2818 . . 3  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  A. x  e.  f  x  ~~  X ) )
4645reximdva 2874 . 2  |-  ( om  ~<_  X  ->  ( E. f  e.  ( UFil `  X ) { y  e.  ~P X  | 
( X  \  y
)  ~<  X }  C_  f  ->  E. f  e.  (
UFil `  X ) A. x  e.  f  x  ~~  X ) )
478, 46mpd 15 1  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) A. x  e.  f  x  ~~  X
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   A.wral 2749   E.wrex 2750   {crab 2753   _Vcvv 3057    \ cdif 3413    C_ wss 3416   ~Pcpw 3963   class class class wbr 4416   dom cdm 4853   ` cfv 5601   omcom 6719    ~~ cen 7592    ~<_ cdom 7593    ~< csdm 7594   cardccrd 8395   fBascfbas 19007   Filcfil 20909   UFilcufil 20963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-ac2 8919
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-rpss 6598  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fi 7951  df-oi 8051  df-card 8399  df-ac 8573  df-cda 8624  df-fbas 19016  df-fg 19017  df-fil 20910  df-ufil 20965
This theorem is referenced by: (None)
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