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Theorem ufilen 17954
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 7107 . . . . . 6  |-  Rel  ~<_
21brrelex2i 4911 . . . . 5  |-  ( om  ~<_  X  ->  X  e.  _V )
3 numth3 8342 . . . . 5  |-  ( X  e.  _V  ->  X  e.  dom  card )
42, 3syl 16 . . . 4  |-  ( om  ~<_  X  ->  X  e.  dom  card )
5 csdfil 17918 . . . 4  |-  ( ( X  e.  dom  card  /\ 
om  ~<_  X )  ->  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  e.  ( Fil `  X
) )
64, 5mpancom 651 . . 3  |-  ( om  ~<_  X  ->  { y  e.  ~P X  |  ( X  \  y ) 
~<  X }  e.  ( Fil `  X ) )
7 filssufil 17936 . . 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 16 . 2  |-  ( om  ~<_  X  ->  E. f  e.  ( UFil `  X
) { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f
)
9 elfvex 5750 . . . . . . 7  |-  ( f  e.  ( UFil `  X
)  ->  X  e.  _V )
109ad2antlr 708 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  X  e.  _V )
11 ufilfil 17928 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( Fil `  X ) )
12 filelss 17876 . . . . . . . 8  |-  ( ( f  e.  ( Fil `  X )  /\  x  e.  f )  ->  x  C_  X )
1311, 12sylan 458 . . . . . . 7  |-  ( ( f  e.  ( UFil `  X )  /\  x  e.  f )  ->  x  C_  X )
1413adantll 695 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  C_  X )
15 ssdomg 7145 . . . . . 6  |-  ( X  e.  _V  ->  (
x  C_  X  ->  x  ~<_  X ) )
1610, 14, 15sylc 58 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  x  ~<_  X )
17 filfbas 17872 . . . . . . . . 9  |-  ( f  e.  ( Fil `  X
)  ->  f  e.  ( fBas `  X )
)
1811, 17syl 16 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( fBas `  X )
)
1918adantl 453 . . . . . . 7  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  f  e.  ( fBas `  X
) )
20 fbncp 17863 . . . . . . 7  |-  ( ( f  e.  ( fBas `  X )  /\  x  e.  f )  ->  -.  ( X  \  x
)  e.  f )
2119, 20sylan 458 . . . . . 6  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  -.  ( X  \  x )  e.  f )
22 difss 3466 . . . . . . . . . . . . . 14  |-  ( X 
\  x )  C_  X
23 elpw2g 4355 . . . . . . . . . . . . . 14  |-  ( X  e.  _V  ->  (
( X  \  x
)  e.  ~P X  <->  ( X  \  x ) 
C_  X ) )
2422, 23mpbiri 225 . . . . . . . . . . . . 13  |-  ( X  e.  _V  ->  ( X  \  x )  e. 
~P X )
25243ad2ant1 978 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
~P X )
26 simp2 958 . . . . . . . . . . . . . 14  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  C_  X )
27 dfss4 3567 . . . . . . . . . . . . . 14  |-  ( x 
C_  X  <->  ( X  \  ( X  \  x
) )  =  x )
2826, 27sylib 189 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  =  x )
29 simp3 959 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  x  ~<  X )
3028, 29eqbrtrd 4224 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  ~<  X )
31 difeq2 3451 . . . . . . . . . . . . . 14  |-  ( y  =  ( X  \  x )  ->  ( X  \  y )  =  ( X  \  ( X  \  x ) ) )
3231breq1d 4214 . . . . . . . . . . . . 13  |-  ( y  =  ( X  \  x )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( X  \  x
) )  ~<  X ) )
3332elrab 3084 . . . . . . . . . . . 12  |-  ( ( X  \  x )  e.  { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  <->  ( ( X  \  x )  e. 
~P X  /\  ( X  \  ( X  \  x ) )  ~<  X ) )
3425, 30, 33sylanbrc 646 . . . . . . . . . . 11  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  x )  e. 
{ y  e.  ~P X  |  ( X  \  y )  ~<  X }
)
35 ssel 3334 . . . . . . . . . . 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 28 . . . . . . . . . 10  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
37363expa 1153 . . . . . . . . 9  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  x  ~<  X )  ->  ( { y  e.  ~P X  | 
( X  \  y
)  ~<  X }  C_  f  ->  ( X  \  x )  e.  f ) )
3837impancom 428 . . . . . . . 8  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  (
x  ~<  X  ->  ( X  \  x )  e.  f ) )
3938con3d 127 . . . . . . 7  |-  ( ( ( X  e.  _V  /\  x  C_  X )  /\  { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f )  ->  ( -.  ( X  \  x
)  e.  f  ->  -.  x  ~<  X ) )
4039impancom 428 . . . . . 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 1183 . . . . 5  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  -.  x  ~<  X ) )
42 bren2 7130 . . . . . 6  |-  ( x 
~~  X  <->  ( x  ~<_  X  /\  -.  x  ~<  X ) )
4342simplbi2 609 . . . . 5  |-  ( x  ~<_  X  ->  ( -.  x  ~<  X  ->  x  ~~  X ) )
4416, 41, 43sylsyld 54 . . . 4  |-  ( ( ( om  ~<_  X  /\  f  e.  ( UFil `  X ) )  /\  x  e.  f )  ->  ( { y  e. 
~P X  |  ( X  \  y ) 
~<  X }  C_  f  ->  x  ~~  X ) )
4544ralrimdva 2788 . . 3  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  A. x  e.  f  x  ~~  X ) )
4645reximdva 2810 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   ~Pcpw 3791   class class class wbr 4204   omcom 4837   dom cdm 4870   ` cfv 5446    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   cardccrd 7814   fBascfbas 16681   Filcfil 17869   UFilcufil 17923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-ac2 8335
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-rpss 6514  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-oi 7471  df-card 7818  df-ac 7989  df-cda 8040  df-fbas 16691  df-fg 16692  df-fil 17870  df-ufil 17925
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