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Theorem ufilen 17883
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 7051 . . . . . 6  |-  Rel  ~<_
21brrelex2i 4859 . . . . 5  |-  ( om  ~<_  X  ->  X  e.  _V )
3 numth3 8283 . . . . 5  |-  ( X  e.  _V  ->  X  e.  dom  card )
42, 3syl 16 . . . 4  |-  ( om  ~<_  X  ->  X  e.  dom  card )
5 csdfil 17847 . . . 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 17865 . . 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 5698 . . . . . . 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 17857 . . . . . . . 8  |-  ( f  e.  ( UFil `  X
)  ->  f  e.  ( Fil `  X ) )
12 filelss 17805 . . . . . . . 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 7089 . . . . . 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 17801 . . . . . . . . 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 17792 . . . . . . 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 3417 . . . . . . . . . . . . . 14  |-  ( X 
\  x )  C_  X
23 elpw2g 4304 . . . . . . . . . . . . . 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 3518 . . . . . . . . . . . . . 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 4173 . . . . . . . . . . . 12  |-  ( ( X  e.  _V  /\  x  C_  X  /\  x  ~<  X )  ->  ( X  \  ( X  \  x ) )  ~<  X )
31 difeq2 3402 . . . . . . . . . . . . . 14  |-  ( y  =  ( X  \  x )  ->  ( X  \  y )  =  ( X  \  ( X  \  x ) ) )
3231breq1d 4163 . . . . . . . . . . . . 13  |-  ( y  =  ( X  \  x )  ->  (
( X  \  y
)  ~<  X  <->  ( X  \  ( X  \  x
) )  ~<  X ) )
3332elrab 3035 . . . . . . . . . . . 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 3285 . . . . . . . . . . 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 7074 . . . . . 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 2739 . . 3  |-  ( ( om  ~<_  X  /\  f  e.  ( UFil `  X
) )  ->  ( { y  e.  ~P X  |  ( X  \  y )  ~<  X }  C_  f  ->  A. x  e.  f  x  ~~  X ) )
4645reximdva 2761 . 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 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   {crab 2653   _Vcvv 2899    \ cdif 3260    C_ wss 3263   ~Pcpw 3742   class class class wbr 4153   omcom 4785   dom cdm 4818   ` cfv 5394    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044   cardccrd 7755   fBascfbas 16615   Filcfil 17798   UFilcufil 17852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-ac2 8276
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-rpss 6458  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-oi 7412  df-card 7759  df-ac 7930  df-cda 7981  df-fbas 16623  df-fg 16624  df-fil 17799  df-ufil 17854
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